Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

TL;DR

This paper introduces a rigorous, spectral, computer-assisted method using Fourier series and Newton-Kantorovich techniques to prove the existence of solutions to semi-linear PDEs on unbounded domains, including applications to solitons.

Contribution

It develops a fully spectral, computer-assisted framework for proving existence of solutions to semi-linear PDEs on unbounded domains, extending to systems with parameters.

Findings

Proved existence of a traveling wave (soliton) in the Kawahara equation.

Constructed approximate inverses of differential operators via Fourier series.

Established stability of the traveling wave solution.

Abstract

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.

Tables

Table 1. Table 1: First three eigenvalues of 𝕃 + D ​ 𝔾 ​ ( u ~ ) 𝕃 𝐷 𝔾 ~ 𝑢 \mathbb{L}+D\mathbb{G}(\tilde{u}) .

Eigenvalue	Value	Radius of the ball of contraction ($R_i$)
${\nu }_1$	-1.145218189524111	$2.8\times {10}^{-11}$
${\nu }_2$	0	$5.7\times {10}^{-11}$
${\nu }_3$	0.8408132697971273	$4.4\times {10}^{-10}$

Equations

$$\mathbb{F}(u)=0$$

$$\Vert D\mathbb{F}(u_0)u{\Vert }_{L^2(\Omega )}\ge |\lambda |\Vert u{\Vert }_{L^2(\Omega )}$$

$$\Vert f{\Vert }_2=\Vert \hat{f}{\Vert }_2,\text{for all }f\in L^2.$$

$$\mathbb{L}u+\mathbb{G}(u)=\psi$$

$$\mathbb{G}(u)\stackrel{\text{mbox}[}{=}\sum _{i=2}^{N_{\mathbb{G}}}{\mathbb{G}}_i(u)$$

$${\mathbb{G}}_i(u)\stackrel{\text{mbox}[}{=}\sum _{k\in J_i}({\mathbb{G}}_{i,k}^{1}u)\cdots ({\mathbb{G}}_{i,k}^{i}u)$$

$$\mathbb{F}(u)\stackrel{\text{mbox}[}{=}\mathbb{L}u+\mathbb{G}(u)-\psi =0,$$

$$\mathbb{L}=D\mathbb{F}(0).$$

$$\mathcal{F}\big{(}\mathbb{L}u\big{)}(\xi)=l(\xi)\hat{u}(\xi),\quad\text{for all }u\in\mathcal{S},$$

$$|l(\xi )|>0,\text{for all }\xi \in {\mathbb{R}}^m.$$

$$H^l\stackrel{\text{mbox}[}{=}\left\{u\in L^2:{\int }_{{\mathbb{R}}^m}|\hat{u}(\xi ){|}^2|l(\xi ){|}^{2}d\xi <\infty \right\},$$

$$(u,v{)}_l\stackrel{\text{mbox}[}{=}{\int }_{{\mathbb{R}}^m}\hat{u}(\xi )\hat{v}(\xi {)}^{\ast }|l(\xi ){|}^{2}d\xi .$$

$$\Vert u{\Vert }_l=\Vert \mathbb{L}u{\Vert }_2,\text{for all }u\in H^l.$$

$$\mathcal{F}\bigg{(}\mathbb{G}_{i,k}^{p}u\bigg{)}(\xi)=g_{i,k}^{p}(\xi)\hat{u}(\xi),\quad\text{for all }\xi\in\mathbb{R}^{m}$$

$$\frac{g_{i,k}^p}{l}\in L^1.$$

$$\mathbb{G}(u)\in H^1({\mathbb{R}}^m),\text{for all }u\in H^l.$$

$${\kappa }_i\ge \sum _{k\in J_i}\underset{p\in \{1,\dots ,i\}}{\min }\Vert \frac{g_{i,k}^1}{l}{\Vert }_2\cdots \Vert \frac{g_{i,k}^{p-1}}{l}{\Vert }_2\Vert \frac{g_{i,k}^p}{l}{\Vert }_{\infty }\Vert \frac{g_{i,k}^{p+1}}{l}{\Vert }_2\cdots \Vert \frac{g_{i,k}^i}{l}{\Vert }_2.$$

$$\Vert {\mathbb{G}}_i(u){\Vert }_2\le {\kappa }_i\Vert u{\Vert }_l^i.$$

$$\left|(1+|\xi |)\mathcal{F}\left(({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u)\right)(\xi )\right|$$

$$\le {\int }_{{\mathbb{R}}^m}|(1+|x|)h_1(x)\hat{u}(x)h_2(\xi -x)\hat{u}(\xi -x)|dx$$

$$+{\int }_{{\mathbb{R}}^m}|h_1(x)\hat{u}(x)(1+|\xi -x|)h_2(\xi -x)\hat{u}(\xi -x)|dx.$$

$$\left|(1+|\xi |)\mathcal{F}\left(({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u)\right)(\xi )\right|$$

$$+\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_2(x)|}{|l(x)|}{\int }_{{\mathbb{R}}^m}|h_1(x)\hat{u}(x)l(\xi -x)\hat{u}(\xi -x)|dx$$

$$=\left(\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_1(x)|}{|l(x)|}\right)(|h_2\hat{u}|\ast |l\hat{u}|)(\xi )$$

$$+\left(\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_2(x)|}{|l(x)|}\right)(|h_1\hat{u}|\ast |l\hat{u}|)(\xi ).$$

$${\Vert (1+|\xi |)\mathcal{F}\left(({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u)\right)(\xi )\Vert }_2$$

$$+\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_2(x)|}{|l(x)|}\Vert h_1\hat{u}{\Vert }_1\Vert l\hat{u}{\Vert }_2$$

$$=\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_1(x)|}{|l(x)|}\Vert h_2\hat{u}{\Vert }_1\Vert u{\Vert }_l$$

$$+\underset{x\in {\mathbb{R}}^m}{\max }\frac{(1+|x|)|h_2(x)|}{|l(x)|}\Vert h_1\hat{u}{\Vert }_1\Vert u{\Vert }_l.$$

$$\Vert h_1\hat{u}{\Vert }_1=\Vert (\frac{h_1}{l})(l\hat{u}){\Vert }_1\le \Vert \frac{h_1}{l}{\Vert }_2\Vert l\hat{u}{\Vert }_2=\Vert \frac{h_1}{l}{\Vert }_2\Vert u{\Vert }_l$$

$${\Vert (1+|\xi |)\mathcal{F}\left(({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u)\right)(\xi )\Vert }_2\le C\Vert u{\Vert }_l^2$$

$$\Vert ({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u){\Vert }_2=\Vert \mathcal{F}(({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u)){\Vert }_2$$

$$=\Vert \frac{h_1}{l}{\Vert }_{\infty }\Vert \frac{h_2}{l}{\Vert }_2\Vert u{\Vert }_l^2.$$

$$\Vert ({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u){\Vert }_2\le \Vert \frac{h_1}{l}{\Vert }_2\Vert \frac{h_2}{l}{\Vert }_{\infty }\Vert u{\Vert }_l^2,$$

$$\Vert ({\mathbb{G}}_{i_1,k_1}^{p_1}u)({\mathbb{G}}_{i_2,k_2}^{p_2}u){\Vert }_2\le \min \left\{\Vert \frac{h_1}{l}{\Vert }_{\infty }\Vert \frac{h_2}{l}{\Vert }_2,\text{leavevmode}\Vert \frac{h_1}{l}{\Vert }_2\Vert \frac{h_2}{l}{\Vert }_{\infty }\right\}\Vert u{\Vert }_l^2.$$

Full text

Key words. Semi-linear PDEs, Unbounded domains, Computer-assisted proof, Fourier analysis, Newton-Kantorovich theorem, Stability of traveling waves

1 Introduction

In this paper, we present a novel Fourier spectral method, to prove (constructively) existence of strong solutions for a wide class of semi-linear Partial Differential Equations (PDEs) defined on $\mathbb{R}^{m}$, possibly with extra equations, such as eigenvalue problems. In particular, we develop tools to build an approximate inverse for the linearization of semi-linear PDE operators around an approximate solution. Moreover, the precision of the approximation can be verified rigorously on the computer. We present the method theoretically in a general framework and we provide tools for the numerical computations of the required bounds. More specifically, the class of semi-linear PDEs we consider may be written as

[TABLE]

defined on a domain $\Omega\subset\mathbb{R}^{m}$ where $\mathbb{F}:X\to Y$ is defined between Hilbert spaces $X$ and $Y$ with norms $\|\cdot\|_{X}$, $\|\cdot\|_{Y}$, respectively. In particular, we assume that $X$ is a closed subset of a Sobolev space $H^{s}(\Omega)$ and $Y$ is a closed subset of $L^{2}(\Omega)$ (or $H^{s}(\Omega)$ in the weak formulation setting). Moreover, $X$ and $Y$ take into account the boundary conditions and any possible symmetries the problem may possess. This set up is natural if one is looking for strong solutions of a given PDE. The use of computer-assisted proofs is motivated by the fact that constructively proving existence of solutions of semi-linear PDEs is a notoriously nontrivial. While non-constructive existence results for weak and strong solutions are vast and numerous, the results and available techniques are seldom if one is interested in quantitative explicit results regarding the solutions (e.g. their shape, their specific location in the function space, the number of oscillations, etc). This is perhaps not surprising, as PDEs are often nonlinear and their solutions live in infinite dimensional function spaces. As a consequence of these hurdles and with the recent increase in computing power, computer-assisted proofs (CAPs) in nonlinear analysis have recently become a popular tool for the rigorous computation of solutions of nonlinear equations. From the early pioneering work on the Feigenbaum conjectures [1], to the existence of chaos and global attractor in the Lorenz equations [2, 3, 4], to the more recent works on Wright’s conjecture [5], chaos in the Kuramoto-Sivashinsky PDE [6], 3D gyroids patterns in materials [7], periodic orbits in Navier-Stokes [8], blowup in 3D Euler [9] and imploding solutions for 3D compressible fluids [10], the role of CAPs has become increasingly present in the analysis of ordinary differential equations, dynamical systems and PDEs. We refer the interested reader to the book [11] and the survey papers [12, 13, 14, 15] for further details.

In this paper, we consider the domain of the PDE (1) to be $\Omega\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{R}^{m}$ and we choose $X\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}H^{l}\subset H^{s}(\mathbb{R}^{m})$ (see the definition in (8)) and $Y\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}L^{2}(\mathbb{R}^{m})$ as Hilbert spaces of interest. The main objective of this article is, given $u_{0}\in H^{l}$, to compute a rigorous approximate inverse of $D\mathbb{F}(u_{0}):H^{l}\to L^{2}(\mathbb{R}^{m})$. More specifically, that is to construct $\mathbb{A}:L^{2}(\mathbb{R}^{m})\to H^{l}$ which approximates $D\mathbb{F}(u_{0})^{-1}:L^{2}(\mathbb{R}^{m})\to H^{l}$ such that an upper bound for the $H^{l}$ operator norm $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$ can be computed explicitly. In particular, if $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}<1$ and if $I_{d}-\mathbb{A}D\mathbb{F}(u_{0})$ is compact, then using a Neumann series argument, we have that $D\mathbb{F}(u_{0}):H^{l}\to L^{2}$ is invertible and $\|D\mathbb{F}(u_{0})^{-1}\|_{2}\leq\frac{\|\mathbb{A}\|_{2,l}}{1-\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}}$. In practice, one wants to have an explicit representation of $\mathbb{A}$ from which standard quantities can easily be computed (e.g. norm, spectrum, action on a vector, etc). Therefore, by writing $D\mathbb{F}(u_{0})^{-1}=\mathbb{A}+(I_{d}-\mathbb{A}D\mathbb{F}(u_{0}))D\mathbb{F}(u_{0})^{-1}$, one can then rigorously approximate the action of $D\mathbb{F}(u_{0})^{-1}$. For instance, the spectrum of $D\mathbb{F}(u_{0})^{-1}$ may be approximated from the spectrum of $\mathbb{A}$, which is in general difficult, especially on an unbounded domain. Consequently, in order to be able to obtain a representation of $\mathbb{A}$ in the computer and use rigorous numerics tools to control its action, we require the construction of $\mathbb{A}$ to be based on Fourier series. In other words, we want to possess a Fourier series representation of $\mathbb{A}$ from which the quantity $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$ can be evaluated through norm computations on Fourier series representations.

In fact, our main motivation for building such an approximate inverse $\mathbb{A}$ is the construction of a fixed point operator. Going back to the general setting of (1) with $\mathbb{F}:X\to Y$, if $u_{0}\in X$ is an approximate solution of (1), then one can build a Newton-like fixed point operator $\mathbb{T}(u)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}u-\mathbb{A}\mathbb{F}(u)$ such that fixed points of $\mathbb{T}$ correspond to zeros of $\mathbb{F}$. The goal is then to prove that the operator $\mathbb{T}$ is well defined, and to find some $r>0$ such that $\mathbb{T}:\overline{B_{r}(u_{0})}\to\overline{B_{r}(u_{0})}$ is a contraction, where $\overline{B_{r}(u_{0})}\subset X$ is the closed ball of radius $r$ centered at $u_{0}\in X$. As a result, the Banach fixed point theorem yields the existence of a unique fixed point of $\mathbb{T}$ in $\overline{B_{r}(u_{0})}$. In particular, if both $u_{0}$ and $\mathbb{A}$ have a rigorous representation in the computer, then one may hope to prove the existence of solutions of (1) using a CAP. In fact, that is why computing a rigorous inverse for $D\mathbb{F}(u_{0})$ is a stepping stone in the field of CAP as it is often the key step in the analysis of such problems. We note, that there are other ways to build the fixed point operator. However, in general, most methods require (at some point) to handle the inverse of the linearization (either in the weak or the strong formulation).

As was touch upon earlier, we note that in order to produce an approximate inverse $\mathbb{A}$, a subsidiary problem is the rigorous representation of functions on $\mathbb{R}^{m}$ in the computer (at least the numerical approximation $u_{0}$) using Fourier series. This has partially been done by Plum et al. [11, 16] by building functions on $H^{1}(\Omega)$ where $\Omega$ is unbounded. In this work, the authors consider $\Omega_{0}\subset\Omega$ a bounded domain and define $\tilde{u}_{0}\in H_{0}^{1}(\Omega_{0})$. Then, using a zero extension of $\tilde{u}_{0}$ outside of $\Omega_{0}$, they obtain a function $u_{0}\in H^{1}(\Omega)$. In particular, they used finite elements in [16] and a sine Fourier series in [11] to build $\tilde{u}_{0}$. This approach results in a representation of $u_{0}$ in the computer. Having this construction in mind, we aim to generalize this process in order to be able to build functions in $H^{s}(\mathbb{R}^{m})$ using only Fourier series. Specifically, given a hypercube $\Omega_{0}$, we need to be able to build $\tilde{u}_{0}\in H_{0}^{s}(\Omega_{0})$ rigorously. The challenge here is to ensure that numerically the represented function is trace-free. As a result, we developed a finite-dimensional trace theorem enabling us to project a given Fourier series from $H^{s}(\Omega_{0})$ into $H_{0}^{s}(\Omega_{0})$ (see Section 4). The advantage of this approach is its numerical feasibility and compatibility with the CAPs methods developed for periodic solutions (e.g. see [7, 17]). Consequently, we may rigorously build functions in $H^{s}(\mathbb{R}^{m})$ having a Fourier series representation. This is, to the best of our knowledge, a new result in the field of rigorous numerics in dynamics.

Depending on the specific spaces $X$ and $Y$, several results exist in computer-assisted proofs in the study of $D\mathbb{F}(u_{0})^{-1}$. We make a distinction between two main problems. The first one is the case where $X$ and $Y$ are spaces of periodic functions on an hypercube $\Omega$. In that case, the analysis on Fourier series can be used and the PDE is turned into a zero finding problem on sequence spaces. Owing to the conjugacy between Fourier series and Fourier transform, we will study the problem on $\mathbb{R}^{m}$. The second kind of PDEs that we want to highlight is non-periodic problems. In that situation, $X$ and $Y$ are Sobolev spaces on a domain $\Omega$ with non-periodic boundary condition. In this setting, finite elements are have been the main tools to study $D\mathbb{F}(u_{0})^{-1}$. We now present a literature review of both problems.

In the periodic case, the PDE (1) is turned into $F(U)=0$ where $F:\hat{X}\to\hat{Y}$ is the new zero-finding problem in sequence space with $\hat{X},\hat{Y}$ two Banach spaces (on Fourier series). Hence, for an approximate solution $U_{0}\in\hat{X}$, the equivalent problem is to bound the norm of $DF(U_{0})^{-1}$. One of the main ingredient to achieve such a goal is to exploit the fact that the Fréchet derivatives are (asymptotically) diagonally dominant (see [7, 8, 18, 19, 20, 21] for some illustrations). Indeed, in semi-linear PDEs, the linear part $DF(0)$ dominates for high-order modes. Therefore, the tail of $DF(U_{0})$ can be seen as a diagonally-dominant infinite dimensional matrix. From this, one can approximate the inverse of $DF(U_{0})$ as a finite matrix acting on a finite part of the sequence, and a tail operator (which is diagonal) which acts on the tail of the sequence. Neumann series arguments then give a way to invert $DF(U_{0})$ and approximate rigorously the inverse. An other way to understand this approach is to notice that $DF(U_{0})-DF(0)$ is relatively compact with respect to $DF(0)$. This feature justifies the use of finite matrices to approximate infinite operators. Moreover, since the problem is set up on a bounded domain, the resolvant of $DF(0)$ (in the cases where it makes sense) is compact. It is this crucial point that differentiates our problem to that in the periodic case (but they are still very close due to similarities between Fourier series and Fourier transform as we will show later). In some cases, the diagonally dominant argument is not needed and the inverse can also be computed explicitly. Indeed, if $U_{0}$ is simple enough, it is shown in [22, 23] that one may analytically build the inverse operator and estimate an upper bound. The linearization $DF(U_{0})$ is, in that case, a tridiagonal operator for which the inverse operator can be built thanks to recurrence relations. Using the asymptotics of these sequences, the inverse may then be approximated. Similarly, the approach in [24] uses recurrence relations to approximate inverse of operators with tridiagonally-dominant linear parts.

For bounded domains $\Omega$ with non-periodic boundary conditions, different approaches have been used to establish the norm of $D\mathbb{F}^{-1}(u_{0}):Y\to X$ or its weak formulation. The first method, that we denote as Nakao’s method [25, 26], is based on finite elements in which the inverse of $DF(u_{0})$ is approximated by the inverse of a finite-dimensional projection on a finite element space. Under certain assumptions on the finite elements space, they are able to invert the linearization and obtain a CAP technique. In fact, this method has recently been improved using the Schur complement [27]. Additionally, Watanabe et al. extended this result and developed specific tools for the inversion of $D\mathbb{F}(u_{0})$ in the elliptic case. Then, using a finite dimensional subspace of Sobolev spaces, they obtained explicit invertibility criteria and the value of the upper bound. For instance, the method presented in [28] gives rigorous estimates for our desired upper bound. The works [29, 30, 31, 32] present further related results extending the method and give sharper criteria for which the inverse can be defined. Also, on bounded domains, Liu et al. [33, 34, 35, 36] developed a method to rigorously estimate eigenvalues of the linearized operator. Using a finite dimensional subspace again, the eigenvalues of the linearized operator are bounded, from above and below, using the eigenvalues of a finite-dimensional operator. Then, using the theory of symmetric operators, the norm of $D\mathbb{F}(u_{0})^{-1}$ is determined using the minimal eigenvalue (in amplitude). Indeed, in the case where $D\mathbb{F}(u_{0})$ is symmetric, then we have that

[TABLE]

where $\lambda$ is the eigenvalue of $D\mathbb{F}(u_{0})$ of minimal amplitude. This method is particularly efficient if $D\mathbb{F}(u_{0})$ is self-adjoint itself and it may be extended to a general operator by considering $D\mathbb{F}(u_{0})^{*}D\mathbb{F}(u_{0})$ where $D\mathbb{F}(u_{0})^{*}$ is the ajoint operator. Finally, the method by Plum et al. [11, 37, 38], in a similar fashion to the approaches in [33, 34, 35, 36], uses an approximation of the eigenvalues of $D\mathbb{F}(u_{0})$ to compute the norm of its inverse. Moreover, as $\Omega$ is bounded, they use the compactness of the resolvant to build a basis of eigenfunctions of the resolvant. Now using an homotopy method and the Temple-Lehmann-Goerisch method, they are able to find a lower and upper bound for $|\lambda|$. Finally, using results on equivalence of norms, a bound for the norm of $D\mathbb{F}^{-1}(u_{0}):L^{2}(\Omega)\to H^{2}(\Omega)\bigcap H_{0}^{1}(\Omega)$ is found.

The common ingredient of these methods is the use, at some point, of compactness which either leads to a natural approximation of infinite operators by matrices or to the spectral theorem that enables to evaluate the norm via eigenvalues. However, as we pointed out earlier, in our case, $\Omega=\mathbb{R}^{m}$ is unbounded and these compactness results do not hold. That is a breaking point between the methods presented above and our problem.

To the best of our knowledge, the only general method that can be extended to unbounded domains is the method described in [11]. Indeed, they consider the weak formulation of $\mathbb{F}(u)=0$ and try to build an upper bound for the norm of the inverse of the weak operator associated to $D\mathbb{F}(u_{0})$. This operator is defined via the inner product of the function space. In that setting, they can use an analysis that falls in the set up of the case of a bounded domain. For the homotopy method part, they use an extra homotopy that links the unbounded domain case to the bounded one. In fact, they use a domain decomposition homotopy [39] to link the eigenvalues of a problem defined on a bounded domain $\Omega_{0}$ to the eigenvalues of the full problem. As a result, they may use the coefficients homotopy as in the bounded domain case. This technique allows to prove weak solution to PDEs on unbounded domain as demonstrated in [11] (where Schrödinger’s equation on $\mathbb{R}^{2}$ is treated) or in [16] where the Navier-Stokes equations are considered on an infinite strip with obstacle. It is worth mentioning that constructive existence of solutions of PDEs defined on the line can be obtained using the parameterization method (e.g. see [40, 41, 42]) to set-up a projected boundary value problem that can then be solved using Chebyshev series or splines (e.g. see [43, 44, 45]).

Now, in the context of the present work, we wish to build a rigorous approximate inverse of the strong formulation of the operator. The main advantage is to be able to prove strong solutions to the PDE, as opposed to weak solutions, which is a important from the point of view of PDE theory. In this case, the challenging part comes from the fact that we cannot use the theory of bi-linear forms of the operators as done in [11]. Moreover, this method allows not only to compute an upper bound for the norm of $D\mathbb{F}(u_{0})^{-1}$, but also to approximate its action on functions. These problems are answered in Section 3 of this work and lead to a new method to build rigorous approximate inverses of strong PDE operators on unbounded domains.

As a consequence, the novelty of our approach is the combination of the theory of Sobolev spaces and Fourier series analysis. The presented method is fully spectral and the required bounds in the computer-assisted approach are computed following the approach in [8, 19, 17, 46]. Consequently, after building an approximate solution $u_{0}$ through its Fourier series representation, the central result (see Section 3) of this article enables us to build a rigorous approximate inverse for $D\mathbb{F}(u_{0})$ using bounded linear operator in $\ell^{2}.$ The construction of this approximation depends on the corresponding Fourier series operator $DF(U_{0})$ defined on $\Omega_{0}$. Moreover, we show in Section 3 that the periodic problem “tends” to the unbounded problem at the rate $\mathcal{Z}_{u}$, where this quantity is exponentially decaying with the size of the domain ${\Omega_{0}}$ (cf. Theorem 3.7).

The paper is organized as follows. We begin in Section 2 by establishing the necessary tools and assumptions in order to set up our method. Then we expose in Section 3 the spectral method to compute a rigorous approximate inverse for $D\mathbb{F}(u_{0})$. In Section 4, we show how the construction of the approximate inverse may be used to prove (constructively) existence of strong solutions via a computer-assisted analysis. In Section 5 we show how the method can be generalized to parameter dependent PDEs with extra equations (in particular eigenvalue problems). Finally, Section 6 provides an application of our approach for the Kawahara equation (sometimes called 5th order KdV equation). In particular, we are able to rigorously prove the existence an stability of a soliton solution. All codes to perform the computer-assisted proofs are available at [47].

2 Set-up of the problem

We begin this section by introducing the usual notations for Sobolev spaces that we will use throughout this article. We refer the interested reader to [48] for a complete presentation of such spaces and their usual properties.

2.1 Notations on Sobolev spaces

Let $m\in\mathbb{N}$ and let $\Omega_{0}\subset\mathbb{R}^{m}$ be a bounded Lipschitz domain. Denote by $H^{s}(\Omega_{0})$ the usual Sobolev space of order $s\in\mathbb{R}$ on $\Omega_{0}$ and $H^{s}(\mathbb{R}^{m})$ the one on $\mathbb{R}^{m}$. Moreover, $(\cdot,\cdot)_{H^{s}(\mathbb{R}^{m})}$ and $\|\cdot\|_{H^{s}(\mathbb{R}^{m})}$ represent the usual inner product and norm on $H^{s}(\mathbb{R}^{m})$, while $(\cdot,\cdot)_{H^{s}(\Omega_{0})}$ and $\|\cdot\|_{H^{s}(\Omega_{0})}$ represent the ones on $H^{s}(\Omega_{0})$. Denote by $H^{s}_{0}(\Omega_{0})$ the closure in $H^{s}(\Omega_{0})$ of the space of infinitely differentiable compactly supported functions on $\Omega_{0}$. When $s=0$, we use the Lebesgue notation $L^{2}=L^{2}(\mathbb{R}^{m})$ or $L^{2}(\Omega_{0}).$ More generally, $L^{p}=L^{p}(\mathbb{R}^{m})$ is the usual $p$ Lebesgue space associated to its norm $\|\cdot\|_{p}$. Denote by $\mathcal{S}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\left\{u\in C^{\infty}(\mathbb{R}^{m}):\sup_{x\in\mathbb{R}^{m}}|x^{\beta}\partial^{\alpha}u(x)|<\infty\text{ for all }\beta,\alpha\in\mathbb{N}^{m}\right\}$ the Schwartz space on $\mathbb{R}^{m}$. Denote by $\mathcal{B}(L^{2})$ the space of bounded linear operators on $L^{2}$ and given $\mathbb{B}\in\mathcal{B}(L^{2})$, denote by $\mathbb{B}^{*}$ the adjoint of $\mathbb{B}$ in $L^{2}.$ Moreover, given $z\in\mathbb{C}$, denote by $z^{*}$ the complex conjugate of $z$.

Denote by $\mathcal{F}:L^{2}\to L^{2}$ the Fourier transform operator and write $\mathcal{F}(f)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\hat{f}$, where $\hat{f}$ is defined as $\hat{f}(\xi)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\int_{\mathbb{R}^{m}}f(x)e^{-i2\pi x\cdot\xi}dx$ for all $\xi\in\mathbb{R}^{m}$. Similarly, the inverse Fourier transform operator is expressed as $\mathcal{F}^{-1}$ and defined as $\mathcal{F}^{-1}(f)(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\int_{\mathbb{R}^{m}}f(\xi)e^{i2\pi x\cdot\xi}d\xi$. In particular, recall the classical Plancherel’s identity

[TABLE]

Finally, for $f_{1},f_{2}\in\mathcal{S}$, the continuous convolution of $f_{1}$ and $f_{2}$ is represented by $f_{1}*f_{2}$.

2.2 Formulation of the problem

Given a function $\psi:\mathbb{R}^{m}\to\mathbb{R}$, we consider the following class of semi-linear PDEs on $\mathcal{S}$

[TABLE]

where $\mathbb{L}$ is a linear differential operator and $\mathbb{G}$ is a purely nonlinear ($\mathbb{G}$ does not contain linear operators) polynomial differential operator of order $N_{\mathbb{G}}\in\mathbb{N}$ where $N_{\mathbb{G}}\geq 2$, that is we can decompose it as a finite sum

[TABLE]

where $\mathbb{G}_{i}$ is a sum of monomials of degree $i$ in the components of $u$ and some of its partial derivatives. In particular, for $i\in\{2,\dots,N_{\mathbb{G}}\}$, $\mathbb{G}_{i}$ can be decomposed as follows

[TABLE]

where $J_{i}\subset\mathbb{N}$ is a finite set of indices, and where $\mathbb{G}^{p}_{i,k}$ is a linear differential operator for all $1\leq p\leq i$ and $k\in J_{i}$. Note that $\mathbb{G}(u)$ depends on the derivatives of $u$ but we only write the dependency in $u$ for simplification. We will emphasize the semi-linearity of the equation later on in Assumption 2.

Letting

[TABLE]

the zeros of $\mathbb{F}$ are solutions of (3). Since $\mathbb{G}$ is purely nonlinear, the linear operator is given by

[TABLE]

We complement equation (3) with two assumptions on the operators $\mathbb{L}$ and $\mathbb{G}$, which are natural to achieve our objectives.

We first assume that $\mathbb{L}=D\mathbb{F}(0):\mathcal{S}\to\mathcal{S}$ is invertible. Assuming the invertibility of $\mathbb{L}$ is a strong assumption. Indeed, our objective is to invert $D\mathbb{F}(u_{0})$, for $u_{0}$ being an approximate solution of our problem. Consequently, asking for the invertibility of $D\mathbb{F}(0)$ can seem to be unrelated. However, we justify this assumption later on in Remark 2.4. Moreover, the invertibility of $\mathbb{L}$ is fundamental in the analysis done in Section 3. We translate this hypothesis using the Fourier transform $l$ of $\mathbb{L}$.

Assumption 1**.**

Assume that the Fourier transform of the linear operator $\mathbb{L}$ is given by

[TABLE]

where $l(\cdot)$ is a polynomial in $\xi$. Moreover, assume that

[TABLE]

Using the definition of the polynomial $l$ from Assumption 1 and using (7), define

[TABLE]

which is a Hilbert space that provides a natural domain for the linear operator $\mathbb{L}$, and it is endowed with inner product

[TABLE]

Denote by $\|u\|_{l}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\sqrt{(u,u)_{l}}$ the induced norm on $H^{l}$. By definition of the inner product on $H^{l}$ and by Plancherel’s identity (2), we obtain from (6) that

[TABLE]

Remark 2.1**.**

Note that $H^{l}$ is an abuse of notation not to be confused with the usual Sobolev spaces $H^{s}$. In general, we have $H^{l}\subset H^{s}(\mathbb{R}^{m})$ for some $s>0$ when $H^{l}$ and $H^{s}(\mathbb{R}^{m})$ are seen as sets of funtions. However, the difference lies in the definition of the norm, which is a crucial point in our analysis as $\mathbb{L}:H^{l}\to L^{2}$ is an isometry (cf. (9)). To differentiate these spaces, we explicitly write the dependency in the domain when using a classical Sobolev space (e.g. $H^{s}(\mathbb{R}^{m})$ or $H^{s}({\Omega_{0}})$) and keep $H^{l}$ for the specific Hilbert space defined in (8).

Recall that we assume the PDE (3) to be semi-linear, which helps with the analysis (e.g. see [49] or [50] for examples). In other words, the degree of differentiability in $\mathbb{G}$ needs to be strictly smaller than the one in $\mathbb{L}$. Using the Hilbert space defined in (8), we consider $\psi\in L^{2}$ and look for $u\in H^{l}$ a solution of the PDE (3). This requires introducing an assumption which implies that $\mathbb{G}$ is well-defined on $H^{l}$ and that $\mathbb{G}_{i}$ (see (4)) is a bounded multi-linear differential operator of order $i$ on $H^{l}$. The next Assumption 2 then provides a sufficient condition under which the operator $\mathbb{G}$ is well-defined and smooth on $H^{l}$ (as proven later on in Lemma 2.2). Moreover, it complements the semi-linearity by specifying some requirements on the monomials of $\mathbb{G}$. These requirements will be useful later on for the analysis of Section 3 (see Lemma 3.6).

Assumption 2**.**

For all $2\leq i\leq N_{\mathbb{G}}$, $k\in J_{i}$ and $1\leq p\leq i$, define $g_{i,k}^{p}$ such that

[TABLE]

and assume that

[TABLE]

Assumption 2 is stronger than the usual semi-linearity assumption as we also assume additional decay for each $g_{i,k}^{p}.$ Nonetheless, throughout this paper, we still refer to Assumption 2 as the semi-linearity assumption. Under Assumptions 1 and 2, one can prove that $\mathbb{G}:H^{l}\to H^{1}(\mathbb{R}^{m})$ is well-defined. The fact that $\mathbb{G}$ maps $H^{l}$ to $H^{1}(\mathbb{R}^{m})$ allows to obtain the regularity of the solutions a posteriori (cf. Proposition 2.3). Hence, looking for solutions in $H^{l}$ becomes equivalent to looking for classical solutions. Moreover, we compute explicit constants $\kappa_{i}>0$ such that $\|\mathbb{G}_{i}(u)\|_{2}\leq\kappa_{i}\|u\|_{l}$ for all $u\in H^{l}$, which are used to perform the nonlinear analysis required in our Newton-Kantorovich approach (see Lemma 4.10 and Lemma 6.3 for instance). The next Lemma 2.2 is central in the set-up of this paper as it provides the aforementioned properties for $\mathbb{G}$ and each $\mathbb{G}_{i}$.

Lemma 2.2**.**

Suppose that Assumptions 1 and 2 are satisfied. Then,

[TABLE]

For each $i\in\{2,\dots,N_{\mathbb{G}}\}$, let $\kappa_{i}>0$ satisfying

[TABLE]

Then, for all $u\in H^{l}$,

[TABLE]

Proof.

Let $u\in H^{l}$ and let $2\leq i_{1},i_{2}\leq N_{\mathbb{G}}$, $k_{1}\in J_{i_{1}}$, $k_{2}\in J_{i_{2}}$, $1\leq p_{1}\leq i_{1}$ and $1\leq p_{2}\leq i_{2}$. Recall the notations of Assumption 2, and denote $h_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}g^{p_{1}}_{i_{1},k_{1}}$ and $h_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}g^{p_{2}}_{i_{2},k_{2}}$. Then,

[TABLE]

Now notice that $\frac{h_{1}}{l}\in L^{1}$ by Assumption 2, therefore $\frac{(1+|\cdot|)h_{1}(\cdot)}{l(\cdot)}\in L^{\infty}$ and similarly $\frac{(1+|\cdot|)h_{2}(\cdot)}{l(\cdot)}\leavevmode\nobreak\ \in\leavevmode\nobreak\ L^{\infty}$ as $l,h_{1}$ and $h_{2}$ are smooth (polynomials). Therefore we get

[TABLE]

Note that $\|l\hat{u}\|_{2}=\|u\|_{l}$ by definition of the norm on $H^{l}$, then using Young’s inequality for the convolution we get

[TABLE]

Moreover, since $\frac{h_{1}}{l},\frac{h_{2}}{l}\in L^{1}$ by Assumption 2 and $h_{1},h_{2},l$ are polynomials, we have $\frac{h_{1}}{l},\frac{h_{2}}{l}\in L^{2}$. Therefore,

[TABLE]

by Cauchy-Schwartz inequality. Similarly, $\|h_{2}\hat{u}\|_{1}\leq\|\frac{h_{2}}{l}\|_{2}\|u\|_{l}$. This implies that there exists $C>0$ such that

[TABLE]

for all $u\in H^{l}.$ Now, recall that if $(1+|\cdot|)f(\cdot)\in L^{2}$ then $f\in H^{1}(\mathbb{R}^{m}).$ In particular, it implies that $(\mathbb{G}^{p_{1}}_{i_{1},k_{1}}u)(\mathbb{G}^{p_{2}}_{i_{2},k_{2}}u)\in H^{1}(\mathbb{R}^{m})$ for all $u\in H^{l}.$ Using the same reasoning recursively, we obtain that $\mathbb{G}(u)\in H^{1}(\mathbb{R}^{m})$ for all $u\in H^{l}.$

Similarly, applying Plancherel’s identity (2) first, one can prove that

[TABLE]

Notice that the role of $h_{1}$ and $h_{2}$ can be permuted in the previous inequality. Consequently, we also get that

[TABLE]

which implies that

[TABLE]

Moreover, by induction we get

[TABLE]

for all $u\in H^{l}$ where $\kappa_{i}$ satisfies (11). ∎

Under Assumptions 1 and 2, $\mathbb{L}$ maps $H^{l}$ to $L^{2}$ (cf. Assumption 1) and $\mathbb{G}$ maps $H^{l}$ to $H^{1}(\mathbb{R}^{m})\subset L^{2}$ (cf. Lemma 2.2). Recalling (3), given $\psi\in L^{2}$, $\mathbb{F}$ maps $H^{l}$ to $L^{2}$ and the PDE (5) is well-defined on $H^{l}$. In addition, in the cases where $\psi$ is smooth, the next Proposition 2.3 shows that looking for classical solutions to (5) is actually equivalent to looking for zeros of $\mathbb{F}$ in $H^{l}$.

Proposition 2.3**.**

Assume $\psi\in H^{\infty}(\mathbb{R}^{m})$ and let $u\in H^{l}$ such that $u$ solves (5). Then $u\in H^{\infty}(\mathbb{R}^{m})$ and $u$ is a classical solution of (5).

Proof.

By assumption $\mathbb{F}(u)=0$ and hence

[TABLE]

or equivalently,

[TABLE]

using Assumption 1. Then, using Lemma 2.2, we know that $\mathbb{G}(u)\in H^{1}(\mathbb{R}^{m})$ so $\mathbb{L}^{-1}\mathbb{G}(u)\in H^{l+1}$, where we define

[TABLE]

Moreover, as $\psi\in H^{\infty}(\mathbb{R}^{m})$ by assumption, we have $\mathbb{L}^{-1}\psi\in H^{\infty}(\mathbb{R}^{m})$. Therefore, using (14), it yields that $u\in H^{l+1}$. Repeating this bootstrapping argument we obtain that $u\in H^{\infty}(\mathbb{R}^{m})$ (see [51] for instance). Consequently, $u$ is a classical solution of (5) (using that $H^{\infty}(\mathbb{R}^{m})\subset C^{\infty}(\mathbb{R}^{m})$ by Sobolev embeddings). ∎

We provide some extra notation required for the analysis of this paper. Given $u\in L^{\infty}$, denote by

[TABLE]

the multiplication operator associated to $u$. More specifically, $\mathbb{u}v=uv$ for all $v\in L^{2}.$ Finally, denote by $\|\cdot\|_{l,2}$ the operator norm for any bounded linear operator between the two Hilbert spaces $H^{l}$ and $L^{2}$. Similarly denote by $\|\cdot\|_{l}$, $\|\cdot\|_{2}$ and $\|\cdot\|_{2,l}$ the operator norms for bounded linear operators on $H^{l}\to H^{l}$, $L^{2}\to L^{2}$ and $L^{2}\to H^{l}$ respectively. More generally, for two given Banach spaces $X$ and $Y$, denote by $\|\cdot\|_{X,Y}$ the operator norm for bounded linear operators on $X\to Y$.

Remark 2.4**.**

The above Assumptions 1 and 2 are natural in the theoretical and numerical study of PDEs, the goal being to obtain computer-assisted proofs of solutions in $H^{l}$. In particular, many PDEs, such as the Korteweg-De Vries (KdV), Kawahara and Swift-Hohenberg equations fall under these assumptions. Moreover, for a given $u_{0}\in H^{l}$, Assumption 1 is necessary to have invertibility of $D\mathbb{F}(u_{0})$. Indeed, by semi-linearity (cf. Assumption 2), $D\mathbb{G}(u_{0})$ is relatively compact with respect to $\mathbb{L}$ and using Weyl’s perturbation theory (see [52]), the essential spectrum of $D\mathbb{F}(u_{0})=\mathbb{L}+D\mathbb{G}(u_{0})$ is equal to the essential spectrum of $\mathbb{L}$. Consequently, if there exists $\xi$ such that $l(\xi)=0$, then [math] is in the essential spectrum of both $\mathbb{L}$ and $D\mathbb{F}(u_{0})$. We thus cannot expect $D\mathbb{F}(u_{0}):H^{l}\to L^{2}$ to be invertible in this set-up. This is an intrinsic characteristic of the unboundedness of the domain. Finally, note that Assumptions 1 and 2 are crucial for the approach introduced in the current paper, and relaxing them is work under investigation (see the conclusion in Section 7 for more details).

2.3 Linearization operators

Given a known element $u_{0}\in H^{l}$ (which will be an approximate solution), we look for solutions to (5) of the form $u+u_{0}$, where we solve for the unknown function $u$. We introduce some notations of the linearization around $u_{0}$ that will be used throughout this article. In particular, $D\mathbb{G}(u_{0}):H^{l}\to L^{2}$ denotes the linearization about $u_{0}$ of the nonlinearity of (5). Note that $D\mathbb{G}(u_{0})$ is a linear combination of operators of the form $\mathbb{v}_{\alpha}\partial^{\alpha}$ (composition of a multiplication by $v_{\alpha}$, given in (15), and a differential operator $\partial^{\alpha}$) where $\alpha\in\mathbb{N}^{m}$ and $v_{\alpha}$ is a function in $H^{1}(\mathbb{R}^{m})$ depending on $u_{0}$ (and possibly on its derivatives). In other words, we have

[TABLE]

where $J_{\mathbb{G}}\subset\mathbb{N}^{m}$ is a finite set and for all $\alpha\in J_{\mathbb{G}}$. Moreover, under Assumption 2, one can prove that the multiplication operator $\mathbb{v}_{\alpha}:L^{2}\to L^{2}$ is well-defined for all $\alpha\in J_{\mathbb{G}}$. This result will be useful in the proof of Theorem 3.4 below.

Proposition 2.5**.**

Let $\alpha\in J_{\mathbb{G}}$, then $v_{\alpha}\in L^{\infty}$ (using the notations of (16)). In particular, the multiplication operator $\mathbb{v}_{\alpha}:L^{2}\to L^{2}$ is well-defined.

Proof.

Let $i\in\{2,\dots,N_{\mathbb{G}}\}$, $k\in J_{i}$ and $p\in\{1,\dots,i\}$, then using (13) we have

[TABLE]

for all $u\in H^{l}$. Therefore, by definition of $v_{\alpha}$ in (16), we know that $\widehat{v_{\alpha}}$ is a finite linear combination of functions of the form $g^{p}_{i,k}\hat{u}$ with $u\in H^{l}$. This implies that $\widehat{v_{\alpha}}\in L^{1}$. Finally, using the definition of the Fourier transform, we get

[TABLE]

for all $x\in\mathbb{R}^{m}$. This concludes the proof. ∎

Remark 2.6**.**

Notice from Assumption 2 that $\frac{\xi^{\alpha}}{l(\xi)}\in L^{1}$ for all $\alpha\in J_{\mathbb{G}}$. This remark will be used later on in the proof of Lemma 3.6.

2.4 Periodic Sobolev spaces

The use of Fourier analysis is widespread in both the theoretical and numerical study of PDEs. We refer the interested reader to [53] or [54] for a review of some applications. In particular, periodic Sobolev spaces are efficient tools for our approach. In this section, we introduce the needed notations for our Fourier analysis on periodic Sobolev spaces. In particular, the objects of Sections 2.2 and 2.3 in $H^{l}$ will have their corresponding representation in periodic Sobolev spaces.

We consider an open cube $\Omega_{0}$ given by

[TABLE]

Moreover, to simplify the analysis, we assume that

[TABLE]

Note that non-uniform hypercubes can be treated in the same fashion. We begin by introducing the tilde notation for indices that will be regularly used throughout this paper.

Definition 2.7**.**

Given $n\in\mathbb{Z}^{m}$, define $\tilde{n}\in\mathbb{Z}^{m}$ as $\tilde{n}_{i}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{n_{i}}{2d_{i}}$ for all $1\leq i\leq m$.

Given $s\geq 0$, denote the Hilbert space

[TABLE]

with inner product

[TABLE]

The space $X^{s}$ which will be used as spaces for the Fourier series coefficients. Similarly as in the continuous case, define the Hilbert space $X^{l}$ as

[TABLE]

with inner product

[TABLE]

Denote by $\|U\|_{l}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\sqrt{(U,U)_{l}}$ the induced norm on $X^{l}$. Finally, $\ell^{p}$ denotes the usual $p$ Lebesgue space for sequences associated to its natural norm $\|\cdot\|_{p}$.

Since elements in $X^{s}$ (respectively $X^{l}$) have a periodic function representation, we denote by $H^{s}_{per}(\Omega_{0})$ (similarly $H^{l}_{per}(\Omega_{0})$) the representation in function space of the elements of $X^{s}$ (respectively $X^{l}$). We use capital letters for sequences $U$ and lowercase letters for functions $u$.

As in the continuous case, we introduce the notion of a convolution, which in this context is the discrete convolution between sequences $U=(u_{n})_{n}$ and $V=(v_{n})_{n}$, and is given by

[TABLE]

The symbol “$*$” is used for both discrete and continuous convolutions when no confusion arises.

Now, our operators in (5) have a representation in $X^{l}$ for the corresponding periodic boundary value problem on $\Omega_{0}$. In fact the linear part $\mathbb{L}$ becomes an operator ${L}$ defined as follows

[TABLE]

By definition of the inner product on $X^{l}$, we obtain that

[TABLE]

Similarly, the nonlinearity $\mathbb{G}$ given in (4) has a representation ${G}$ in $X^{l}$ that can be written as

[TABLE]

where ${G}_{i}$ is a multi-linear operator on $X^{l}$ of order $i$, and where products are interpreted as discrete convolutions. Hence, we can define $F:X^{l}\to\ell^{2}$ and the equivalent of the zero finding problem (5) on $X^{l}$ as

[TABLE]

where $\Psi$ is the Fourier series representation in $\ell^{2}$ of $\psi|_{\Omega_{0}}$ (the restriction of $\psi$ to $\Omega_{0}$). Similarly as in the continuous case, given $U=(u_{n})_{n\in\mathbb{Z}^{m}}\in\ell^{1}$,

[TABLE]

is the discrete convolution operator associated to $U$, that is $(\mathbb{U}V)_{n}=\displaystyle\sum_{k\in\mathbb{Z}^{m}}u_{k}v_{n-k}$ for all $n\in\mathbb{Z}^{m}$ and all $V=(v_{n})_{n\in\mathbb{Z}^{m}}\in\ell^{2}.$ Note that $U*V\in\ell^{2}$ for all $U\in\ell^{1}$ and all $V\in\ell^{2}$ using Young’s inequality for the convolution. Finally, for $\alpha\in J_{\mathbb{G}}$, we slightly abuse notation and denote by $\partial^{\alpha}$ the linear operator given by

[TABLE]

In other words, given $U\in X^{l}$ and $u_{per}\in H^{l}_{per}(\Omega_{0})$ its function representation on ${\Omega_{0}}$, then $\partial^{\alpha}U$ is the Fourier series of $\partial^{\alpha}u_{per}.$

3 Approximation of the inverse of $D\mathbb{F}(u_{0})$

The main difficulty in the study of solutions of (5) on $\mathbb{R}^{m}$ is that despite the invertibility of $\mathbb{L}$, the operator $\mathbb{L}^{-1}$ is not compact, as it would be the case on a bounded domain. Compactness is often used when one needs to numerically approximate operators in a PDE. In fact, it is a central ingredient in computer-assisted proofs based on spectral methods, as it justifies approximating infinite-dimensional operators by matrices. Hence, this leads to an important difference with the study of PDEs on unbounded domains. From that perspective, our challenge is to find a way to gain back some notion of compactness. This is achieved by considering $D\mathbb{G}(u_{0})\mathbb{L}^{-1}$ instead of directly $\mathbb{L}^{-1}$. In this section we fix some $u_{0}\in H^{l}$ such that supp$(u_{0})\subset\Omega_{0}$, representing an approximate solution of (5).

3.1 Preliminary results

Given $u_{0}\in H^{l}$ with supp$(u_{0})\subset\Omega_{0}$, we develop a method to approximate $D\mathbb{F}(u_{0})^{-1}\leavevmode\nobreak\ :\leavevmode\nobreak\ L^{2}\leavevmode\nobreak\ \to\leavevmode\nobreak\ H^{l}$. Recall that Assumption 1 provides that $\mathbb{L}:H^{l}\to L^{2}$ has a continuous inverse $\mathbb{L}^{-1}:L^{2}\to H^{l}$. Using this property, we can prove the following result.

Lemma 3.1**.**

Let $w_{0}\in H^{l}$, then $D\mathbb{G}(w_{0})\mathbb{L}^{-1}\colon L^{2}\to L^{2}$ is compact.

Proof.

Suppose first that supp$(w_{0})\subset B_{r}$ for some $r>0$ where $B_{r}$ is the open ball centered at [math] in $\mathbb{R}^{m}$. Let $u\in H^{l}$, then by Lemma 2.2, we have that $D\mathbb{G}(w_{0})u\in H^{1}(\mathbb{R}^{m})$. Moreover supp$(D\mathbb{G}(w_{0})u)\subset B_{r}$. Let $v$ be the restriction of $D\mathbb{G}(w_{0})u$ on $B_{r}$. By construction, we obtain $v\in H^{1}_{0}(B_{r})$. From the Rellich-Kondrachov theorem (see [48]), we obtain that $H^{1}_{0}(B_{r})$ is compactly embedded in $L^{2}(B_{r})$. As a consequence it yields that $D\mathbb{G}(w_{0})\mathbb{L}^{-1}$ is a compact operator from $L^{2}$ to itself. Then using that the set of smooth functions with compact support is dense in $H^{l}$ (see [51]) and that the set of compact operators is closed, we obtain the proof for all $w_{0}\in H^{l}$. ∎

Knowing that $D\mathbb{G}(u_{0})\mathbb{L}^{-1}:L^{2}\to L^{2}$ is compact, we can approximate it by finite range operators. This justifies the construction we are about to present. We want to build $\mathbb{A}:L^{2}\to H^{l}$ such that $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}<1$ is small. First, notice that $\mathbb{A}=\mathbb{L}^{-1}\mathbb{L}\mathbb{A}=\mathbb{L}^{-1}\mathbb{B}$ with

[TABLE]

Therefore, equivalently, one can look for $\mathbb{B}:L^{2}\to L^{2}$ approximating the inverse of $D\mathbb{F}(u_{0})\mathbb{L}^{-1}=I_{d}+D\mathbb{G}(u_{0})\mathbb{L}^{-1}:L^{2}\to L^{2}.$ In particular, notice that $I_{d}+D\mathbb{G}(u_{0})\mathbb{L}^{-1}:L^{2}\to L^{2}$ is a compact perturbation of the identity (using Lemma 3.1), therefore, we look for $\mathbb{B}_{\Omega_{0}}$ such that

[TABLE]

where $\mathbb{B}_{\Omega_{0}}:L^{2}\to L^{2}$ is a compact operator to be determined. Moreover, we assume that

[TABLE]

Under this assumption, one can prove that $\mathbb{B}_{\Omega_{0}}$ can be represented by a bounded linear operator $B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$ on Fourier series. The next Lemma 3.2 provides the construction of $B_{\Omega_{0}}.$

Lemma 3.2**.**

Let $\mathbb{B}_{\Omega_{0}}:L^{2}\to L^{2}$ be a bounded linear operator such that $\mathbb{B}_{\Omega_{0}}=\mathbb{1}_{\Omega_{0}}\mathbb{B}_{\Omega_{0}}\mathbb{1}_{\Omega_{0}}$. Then $\mathbb{B}_{\Omega_{0}}$ has a unique representation as a Fourier series operator $B_{\Omega_{0}}=(b_{p,j})_{p,j\in\mathbb{Z}^{m}}:\ell^{2}\to\ell^{2}$ given by

[TABLE]

where $\psi_{n}(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}e^{i2\pi\tilde{n}\cdot x}$ for all $n\in\mathbb{Z}^{m}$ and all $x\in{\Omega_{0}}.$ Moreover, we abuse notation and for a given operator $B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$, we denote by

[TABLE]

its associated operator $\mathbb{B}_{\Omega_{0}}:L^{2}\to L^{2}$ such that $\mathbb{B}_{\Omega_{0}}=\mathbb{1}_{\Omega_{0}}\mathbb{B}_{\Omega_{0}}\mathbb{1}_{\Omega_{0}}$. Finally, we have

[TABLE]

Proof.

The construction of $B_{\Omega_{0}}$ is a direct consequence of the fact that Fourier series form a basis for $L^{2}({\Omega_{0}}).$ The norm equality $\|\mathbb{B}_{\Omega_{0}}\|_{2}=\|B_{\Omega_{0}}\|_{2}$ follows immediately from Parseval’s identity. ∎

Let us provide some intuition on the construction of the previous lemma. Given a Banach space $H$ of functions defined on $\mathbb{R}^{m}$, denote by $H_{\Omega_{0}}\subset H$ the subspace of $H$ defined as

[TABLE]

For instance, we have $L^{2}_{\Omega_{0}}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\{u\in L^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\ \text{supp}(u)\subset{\Omega_{0}}\}$ or $H_{\Omega_{0}}^{l}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\{u\in H^{l}\leavevmode\nobreak\ :\leavevmode\nobreak\ \text{supp}(u)\subset{\Omega_{0}}\}$. In particular, given $u\in L^{2}_{\Omega_{0}}$ (resp. $u\in H^{l}_{\Omega_{0}}$) we have that $u$ has a Fourier series representation in $\ell^{2}$ (resp. $X^{l}$) on ${\Omega_{0}}$. In other words, there exists $U=(u_{n})_{n\in\mathbb{Z}^{m}}\in\ell^{2}$ (resp. $U\in X^{l}$) such that

[TABLE]

Moreover we have

[TABLE]

Therefore, for each $u\in L^{2}_{\Omega_{0}}$ (resp. $u\in H^{l}_{\Omega_{0}}$) we associate to it its Fourier series series representation $U\in\ell^{2}$ (resp. $U\in X^{l}$).

Recall that $\mathcal{B}(L^{2})$ denotes the space of bounded linear operators on $L^{2}$ and denote by $\mathcal{B}_{\Omega_{0}}(L^{2})$ the following subspace of $\mathcal{B}(L^{2})$

[TABLE]

Now, let $u\in L^{2}$, then by construction $\mathbb{1}_{\Omega_{0}}u\in L^{2}_{\Omega_{0}}$. Moreover, $\mathbb{1}_{\Omega_{0}}u$ has a Fourier series representation $U=(u_{n})_{n\in\mathbb{Z}^{m}}$ given in (25). Then, if $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(L^{2})$, we have $g\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{B}_{\Omega_{0}}u=\mathbb{1}_{\Omega_{0}}\mathbb{B}_{\Omega_{0}}u$. By construction, $g=\mathbb{1}_{\Omega_{0}}g$ so $g$ also has a Fourier series representation $(g_{n})_{n\in\mathbb{Z}^{m}}$. Finally, Lemma 3.2 gives that $g_{n}$ can be obtained as $g_{n}=(B_{\Omega_{0}}U)_{n}$ for all $n\in\mathbb{Z}^{m}$. Therefore, the action of $\mathbb{B}_{\Omega_{0}}$ on $L^{2}$ is determined by the action of $B_{\Omega_{0}}$ on $\ell^{2}$ and vice versa. Moreover, Parseval’s identity (23) provides a natural norm equality between the two representations of the operator.

Lemma 3.2 is central in our analysis. Indeed, numerically, we build $B=I_{d}+B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$ as an approximate inverse for $I_{d}+DG(U_{0})L^{-1}$. In particular, we can choose $B_{\Omega_{0}}$ to be a matrix so that $B_{\Omega_{0}}$ is compact. Lemma 3.2 provides that $B_{\Omega_{0}}$ has an operator representation $\mathbb{B}_{\Omega_{0}}=\mathcal{F}^{-1}(B_{\Omega_{0}}):L^{2}\to L^{2}$ (which is compact as well by construction) such that $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(L^{2}).$ Finally, we build $\mathbb{A}=\mathbb{L}^{-1}(I_{d}+\mathbb{B}_{\Omega_{0}}):L^{2}\to H^{l}.$ Therefore, $\mathbb{A}$ is completely determined by $B$, which is a numerical approximation of $I_{d}+DG(U_{0})L^{-1}.$ Moreover, if the size of ${\Omega_{0}}$ is big enough, $\mathbb{A}$ accurately approximates $D\mathbb{F}(u_{0})^{-1}$ (this point is clarified in Theorem 3.4 below). In addition, the matrix $B_{\Omega_{0}}$ can be constructed numerically in intervals (using the package IntervalArithmetic.jl [55] for instance). Consequently, the computations associated to $\mathbb{A}$, such as norms, spectrum, etc, can be obtained rigorously using interval arithmetic [56]. As an illustration, the next lemma provides the operator norm computation of $\mathbb{A}$ using the norm of $B$.

Lemma 3.3**.**

Suppose that $\mathbb{A}=\mathbb{L}^{-1}\mathbb{B}$ with $\mathbb{B}=I_{d}+\mathbb{B}_{\Omega_{0}}$ and $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}$. Moreover, denote by $B_{\Omega_{0}}$ the Fourier series representation of $\mathbb{B}_{\Omega_{0}}$ (given in Lemma 3.2) and define $B=I_{d}+B_{\Omega_{0}}$. Then,

[TABLE]

Proof.

Using (9), we get

[TABLE]

Let $u\in L^{2}$. By construction $\mathbb{B}=I_{d}+\mathbb{B}_{\Omega_{0}}$ where $\mathbb{B}_{\Omega_{0}}=\mathbb{1}_{\Omega_{0}}\mathbb{B}_{\Omega_{0}}\mathbb{1}_{\Omega_{0}}$ and therefore

[TABLE]

But notice that

[TABLE]

and similarly,

[TABLE]

Therefore,

[TABLE]

Finally, we use Parseval’s identity (23) to get $\|\mathbb{1}_{\Omega_{0}}(I_{d}+\mathbb{B}_{\Omega_{0}})\|_{2}=\|B\|_{2}$. ∎

3.2 Upper bounds for $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$ and $\|D\mathbb{F}(u_{0})^{-1}\|_{2,l}$

Using the construction of the previous Section 3.1, we build an approximate inverse $\mathbb{A}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{L}^{-1}(I_{d}+\mathbb{B}_{\Omega_{0}}):L^{2}\to H^{l}$ for $D\mathbb{F}(u_{0})$. More specifically, $\mathbb{B}_{\Omega_{0}}$ has a Fourier series representation $B_{\Omega_{0}}$ such that $I_{d}+B_{\Omega_{0}}$ approximates the inverse of $I_{d}+DG(U_{0})L^{-1}$. Given $\mathbb{A}$ constructed in such a way, the following Theorem 3.4 provides an upper bound for $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$. In other words, it measures how well $\mathbb{A}$ approximates $D\mathbb{F}(u_{0})^{-1}$. Moreover, under the assumption that $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}<1$, Theorem 3.4 provides an upper bound for $\|D\mathbb{F}(u_{0})^{-1}\|_{2,l}$.

Before stating Theorem 3.4, the analysis requires some additional notations. Indeed, we want a rigorous upper bound for $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$ on the computer, which requires to approximate rigorously $D\mathbb{F}(u_{0})=\mathbb{L}+D\mathbb{G}(u_{0})$. To do so, we approximate instead $D\mathbb{F}(u_{0})\mathbb{L}^{-1}=I_{d}+D\mathbb{G}(u_{0})\mathbb{L}^{-1}$. The operator $D\mathbb{G}(u_{0})\mathbb{L}^{-1}:L^{2}\to L^{2}$ being compact (see Lemma 3.1), it can be approximated by finite range operators, meaning that a matrix approximation on a computer can be justified and quantified. Now, using (16), note that

[TABLE]

In particular, since $u_{0}\in H^{l}_{\Omega_{0}}$, then for all $\alpha\in J_{\mathbb{G}}$

[TABLE]

In practice, we consider the adjoint in $L^{2}$ of this operator as it leads to a more natural analysis. We have

[TABLE]

as $\partial^{\alpha}$ and $\mathbb{L}^{-1}$ commute. For each $\alpha\in J_{\mathbb{G}}$, using the notation of (22), let

[TABLE]

the bounded linear operator such that $\mathbb{K}_{\alpha}\in\mathcal{B}_{\Omega_{0}}(L^{2})$ and $\mathbb{K}_{\alpha}$ has a Fourier series representation given by $(\partial^{\alpha}L^{-1})^{*}$ (the construction is justified by Lemma 3.2). In particular, $\mathbb{K}_{\alpha}\mathbb{v}_{\alpha}^{*}$ approximates $(\partial^{\alpha}\mathbb{L}^{-1})^{*}\mathbb{v}_{\alpha}^{*}$ in $\mathcal{B}(L^{2})$.

Using the construction of $\mathbb{K}_{\alpha}$, we derive in the next result an upper bound $\mathcal{Z}_{1}$ for the operator norm $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}$. Note that $\mathcal{Z}_{1}$ is the sum of two components, namely a bound $Z_{1}$ which depends on the periodic problem only and measures how well $A=L^{-1}(I_{d}+B_{\Omega_{0}})$ approximates the inverse for $DF(U_{0})$; and a bound $\mathcal{Z}_{u}$ which arises from the unboundedness of the domain and which depends on the approximation of the operator $(\partial^{\alpha}\mathbb{L}^{-1})^{*}\mathbb{v}_{\alpha}^{*}$ by the operator $\mathbb{K}_{\alpha}\mathbb{v}_{\alpha}^{*}$.

Theorem 3.4**.**

Let $\mathbb{A}:L^{2}\to H^{l}$ be a bounded linear operator such that $\mathbb{A}=\mathbb{L}^{-1}\mathbb{B}$ where $\mathbb{B}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}I_{d}+\mathbb{B}_{\Omega_{0}}$, with $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(L^{2})$. Let $B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$ be its Fourier series representation given in Lemma 3.2. Finally, denote $B=I_{d}+B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$ and $A=L^{-1}B:\ell^{2}\to X^{l}$. Assume that the bounds $Z_{1}$ and $\mathcal{Z}_{u}$ satisfy

[TABLE]

Then, defining $\mathcal{Z}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}Z_{1}+\mathcal{Z}_{u}$, we have

[TABLE]

Moreover, if $\mathcal{Z}_{1}<1$, then both $\mathbb{A}:L^{2}\to H^{l}$ and $D\mathbb{F}(u_{0}):H^{l}\to L^{2}$ have a bounded inverse and

[TABLE]

Proof.

First, recall that $\mathbb{B}=\mathbb{L}\mathbb{A}$, then using (9) and (28) we get

[TABLE]

where the last inequality follows from triangle’s inequality and where $\mathbb{K}_{\alpha}$ is defined in (30).

Now, recall that $\mathbb{B}=I_{d}+\mathbb{B}_{\Omega_{0}}$, therefore

[TABLE]

However, notice that

[TABLE]

as $\mathbb{B}_{\Omega_{0}},\mathbb{K}_{\alpha}\in\mathcal{B}_{\Omega_{0}}(L^{2})$ by assumption. Then using Proposition 2.5 and (29), we obtain that $\mathbb{v}_{\alpha}\in\mathcal{B}_{\Omega_{0}}(L^{2})$ and it has a Fourier series representation given by $\mathbb{V}_{\alpha}:\ell^{2}\to\ell^{2}.$ Consequently, using Lemma 3.2 and (30), we obtain that $\sum_{\alpha\in J_{\mathbb{G}}}\mathbb{v}_{\alpha}\mathbb{K}_{\alpha}^{*}$ has a Fourier series representation given by $DG(U_{0})L^{-1}.$ Therefore, using Parseval’s identity (23), we get

[TABLE]

as $A=L^{-1}B$ and $\|U\|_{l}=\|LU\|_{2}$ for all $U\in X^{l}.$ Let us now focus on the second term $\|\mathbb{B}\sum_{\alpha\in J_{\mathbb{G}}}\mathbb{v}_{\alpha}\left(\mathbb{K}_{\alpha}^{*}-\partial^{\alpha}\mathbb{L}^{-1}\right)\|_{2}$ in (34). Recall that for a bounded linear operator $\mathbb{T}:L^{2}\to L^{2}$, then $\|\mathbb{T}\|_{2}=\|\mathbb{T}^{*}\|_{2}$ where $\mathbb{T}^{*}$ is the adjoint of $\mathbb{T}.$ Using this identity, we get

[TABLE]

where the last equality follows from formula (27) in Lemma 3.3. We conclude that (32) holds by combining (34), (35) and (36).

Finally, let us compute an upper bound for $\|D\mathbb{F}(u_{0})^{-1}\|_{2,l}$. Let $u\in H^{l}$, then

[TABLE]

If $1-\mathcal{Z}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}1-Z_{1}-\mathcal{Z}_{u}>0$, then

[TABLE]

for all $u\in H^{l}$. But Lemma 3.1 yields that $I_{d}+D\mathbb{G}(u_{0})\mathbb{L}^{-1}$ is a Fredholm operator, therefore, we obtain that $D\mathbb{F}(u_{0})$ is invertible and

[TABLE]

where we used formula (27) from Lemma 3.3. Moreover, using that $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}<1$ combined with a Neumann series argument, we have that $\mathbb{A}D\mathbb{F}(u_{0}):H^{l}\to H^{l}$ has a bounded inverse and therefore $\mathbb{A}:L^{2}\to H^{l}$ has a bounded inverse as well. ∎

Remark 3.5**.**

In the case where $B=I_{d}+B_{\Omega_{0}}$ and $B_{\Omega_{0}}$ is a matrix, the quantity $\|B\|_{2}$ can be evaluated as a matrix norm (see the proof of Lemma 4.10). Moreover, $Z_{1}$ is the usual bound one has to compute for the proof of a periodic solution using the radii polynomial approach (see [8] or [19] for instance), the only difference being that here the bound needs to be computed with the $\ell^{2}$ norm, as opposed to a more standard (weighed) $\ell^{1}$ norm computation. Because we deal with an unbounded domain, the extra term $\mathcal{Z}_{u}$ has to be taken into account. One can expect that $\mathcal{Z}_{u}\to 0$ as $d\to\infty$, which is the result we derive in the next Section 3.3. Furthermore, if one knows that $DF(U_{0})$ is invertible, then one can choose $A=DF(U_{0})^{-1}$ in the previous Theorem 3.4 (so that $Z_{1}=0$). Moreover, the above reasoning provides

[TABLE]

where this time $\mathcal{Z}_{u}>0$ satisfies

[TABLE]

Therefore, if one can compute an upper bound for $\|DF(U_{0})^{-1}\|_{2,l}$ (without necessarily using an approximate inverse $\mathbb{A}$), then the previous lemma provides an upper bound for $D\mathbb{F}(u_{0})^{-1}$ under the condition that $\sum_{\alpha\in J_{\mathbb{G}}}\|\left(\mathbb{K}_{\alpha}-(\partial^{\alpha}\mathbb{L}^{-1})^{*}\right)\mathbb{v}_{\alpha}^{*}\|_{2}$ is sufficiently small. This condition can be verified using Theorem 3.7 below.

3.3 Computation of the bound $\mathcal{Z}_{u}$

In the previous Section 3.2, given an approximate inverse $\mathbb{A}$ defined in Section 3.1, we derived a strategy to compute an upper bound $\mathcal{Z}_{1}$ satisfying $\|I_{d}-\mathbb{A}D\mathbb{F}(u_{0})\|_{l}\leq\mathcal{Z}_{1}=Z_{1}+\mathcal{Z}_{u}$, where both $Z_{1}$ and $\mathcal{Z}_{u}$ are defined in (31). The bound $Z_{1}$ can easily be obtained as its rigorous computation is well-known in the field of computer-assisted proofs in nonlinear analysis (see [19] or [8] for instance). The novelty lies in the bound $\mathcal{Z}_{u}$ and this section focuses on its computation. In particular, we derive a method to compute the quantity $\|\left(\mathbb{K}_{\alpha}-(\partial^{\alpha}\mathbb{L}^{-1})^{*}\right)\mathbb{v}_{\alpha}^{*}\|_{2}$ for a given $\alpha\in J_{\mathbb{G}}$ and where $\mathbb{K}_{\alpha}$ is defined in (30).

First, notice that $(\partial^{\alpha}\mathbb{L}^{-1})^{*}$ has a Fourier transform given by $\xi\to\frac{(-i2\pi\xi)^{\alpha}}{l(-\xi)}$. Let us define $f_{\alpha}:\mathbb{R}^{m}\to\mathbb{R}$ through its Fourier transform as

[TABLE]

for all $\xi,x\in\mathbb{R}^{m}$. Then, fixing $u\in L^{2}$ and $\alpha\in J_{\mathbb{G}}$, we get that

[TABLE]

More specifically, the operator $(\partial^{\alpha}\mathbb{L}^{-1})^{*}$ is a convolution operator by the function $f_{\alpha}$. The next lemma demonstrates that $f_{\alpha}$ is exponentially decaying. This decay is essential in the approximation of $(\partial^{\alpha}\mathbb{L}^{-1})^{*}$ by $\mathbb{K}_{\alpha}.$

Lemma 3.6**.**

Let $\alpha\in J_{\mathbb{G}}$ and recall $f_{\alpha}$ from (37). Then, there exist constants $C_{\alpha}$ and $a>0$ such that

[TABLE]

for all $x\in\mathbb{R}^{m}$ where $|x|_{1}=|x_{1}|+\cdots+|x_{m}|$. Moreover, $a$ is independent of $\alpha$.

Proof.

Fix $\alpha\in J_{\mathbb{G}}$. Remark 2.6 implies that $\frac{(-i2\pi\xi)^{\alpha}}{l(-\xi)}\in L^{1}$. Denote by $\{b_{j}+ia_{j}\}_{j}$ the finite set of roots of the polynomial $l(\cdot)$. Let $\tilde{a}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\min_{j}|a_{j}|$, using Assumption 1, we know that $|l|>0$ so $\tilde{a}>0$. In particular, $\tilde{a}$ only depends on $l$ but not on $\alpha$. Moreover, choosing $0<{a}<\tilde{a}$, Cauchy’s theorem (see Chapter IX in [57]) implies that there exists $C_{\alpha}$ such that (39) holds. ∎

The previous Lemma 3.6 provides an exponential decay for $f_{\alpha}$, a result which we now use to control rigorously the error when approximating $(\partial^{\alpha}\mathbb{L}^{-1})^{*}\mathbb{v}_{\alpha}^{*}$ by $\mathbb{K}_{\alpha}\mathbb{v}_{\alpha}^{*}$. To see this, for each $\alpha\in J_{\mathbb{G}}$, let us first define the following quantities

[TABLE]

In particular, notice that

[TABLE]

where $\mathcal{Z}_{u}$ satifies (31). Given $\alpha\in J_{\mathbb{G}}$, the next Theorem 3.7 provides an upper bound for both $Z^{\alpha}_{u,1}$ and $Z^{\alpha}_{u,2}$. Then, using (42), one can compute a value for $\mathcal{Z}_{u}$.

Theorem 3.7**.**

Let $\alpha\in J_{\mathbb{G}}$ and let $\mathbb{K}_{\alpha}$ as in (30). Recall that $\Omega_{0}=(-d,d)^{m}$, and let $E_{1}$ and $E_{2}$ be the Fourier series representations of the functions $x\mapsto e^{2ad}-e^{2ad}\prod_{i=1}^{m}\left(1-e^{-2ad}\cosh(2ax_{i})\right)$ and $x\mapsto\prod_{i=1}^{m}\cosh(2ax_{i})$ on ${\Omega_{0}}$ respectively. Then,

[TABLE]

where $V_{\alpha}$ is defined in (29). Moreover, there exists $C(d)>0$ depending algebraically on $d$ and independent of $\alpha$, such that

[TABLE]

Proof.

Let $u\in L^{2}$ such that $\|u\|_{2}=1$ and let $v\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}v_{\alpha}^{*}u$. First, recall that

[TABLE]

as $\mathbb{K}_{\alpha}=\mathbb{1}_{\Omega_{0}}\mathbb{K}_{\alpha}$ by construction. We begin by focus our attention on bounding the first term in the right-hand side of (45), that is $\|\mathbb{1}_{\mathbb{R}^{m}\setminus\Omega_{0}}(\partial^{\alpha}\mathbb{L}^{-1})^{*}v\|_{2}$. Recall that

[TABLE]

for all $x\in\mathbb{R}^{m}$ by definition of $f_{\alpha}.$ Then, using Lemma 3.6 and (29), we obtain

[TABLE]

where in the first equality we used that supp$(v_{\alpha})\subset\Omega_{0}$.

Now using Cauchy-Schwartz inequality and Fubini’s theorem, we get

[TABLE]

as $\|u\|_{2}=1$ by assumption. Then, let $x\in{\Omega_{0}}$, we have

[TABLE]

Therefore, combining (3.3) and (3.3), we get

[TABLE]

Recall that by hypothesis the vector $E_{1}$ is the Fourier series representation of the function $x\mapsto\displaystyle e^{2ad}-e^{2ad}\prod_{i=1}^{m}\left(1-e^{-2ad}\cosh(2ax_{i})\right)$ on ${\Omega_{0}}$, and hence we obtain

[TABLE]

using Parseval’s identity. Finally, we conclude that the first term in the right-hand side of (45) is bounded by

[TABLE]

Let us now find an explicit upper bound for the second term in the right-hand side of (45), namely for the term $\|\mathbb{1}_{\Omega_{0}}((\partial^{\alpha}\mathbb{L}^{-1})^{*}-\mathbb{K}_{\alpha})v\|_{2}$. Let $g\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{1}_{\Omega_{0}}((\partial^{\alpha}\mathbb{L}^{-1})^{*}-\mathbb{K}_{\alpha})v$. By construction $g\in L^{2}_{\Omega_{0}}$. Therefore $g$ has a Fourier series representation $(g_{n})_{n\in\mathbb{Z}^{m}}\in\ell^{2}$ defined as

[TABLE]

for all $n\in\mathbb{Z}^{m}$. Let $n\in\mathbb{Z}^{m}$, then notice that

[TABLE]

by definition of $\mathbb{L}$. But as $v=v_{\alpha}u$ has support on $\Omega_{0}$ (since supp$(v_{\alpha})\subset\Omega_{0}$), we have

[TABLE]

where $(v_{n})_{n\in\mathbb{Z}^{m}}$ is the series of Fourier coefficients of $v$. Therefore,

[TABLE]

Now, the Fourier series representation of $\mathbb{K}_{\alpha}\in\mathcal{B}_{\Omega_{0}}(L^{2})$ is $(\partial^{\alpha}L^{-1})^{*}$ by definition, and so

[TABLE]

Therefore, combining (3.3), (50) and (51), we obtain

[TABLE]

Then, denote by ${\Omega_{0}}+2dn\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(-d_{1}+2dn_{1},d_{1}+2dn_{1})\times\cdots\times(-d_{m}+2dn_{m},d_{m}+2dn_{m})$ and using Parseval’s identity, we obtain

[TABLE]

where we used that $\displaystyle\frac{1}{|{\Omega_{0}}|}\sum_{n\in\mathbb{Z}^{m}}e^{i2\pi\tilde{n}\cdot(y-x)}=\sum_{n\in\mathbb{Z}^{m}}\delta(y-x-2dn)$, which is the Dirac comb.

Let $n\in\mathbb{Z}^{m}$, then using (38) and Lemma 3.6 we get

[TABLE]

using Fubini’s theorem.

Then, notice that if $n=0$, we have

[TABLE]

which is the quantity we computed in (48). Moreover, one can prove that there exists a constant $C(d)>0$, depending on $d$ algebraically and independent of $\alpha$, such that

[TABLE]

using some standard integration techniques of exponentials. Therefore, we obtain

[TABLE]

Then, recall that $v\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}v_{\alpha}^{*}u$ and using Cauchy Schwartz inequality, we get

[TABLE]

Finally, using that $\cosh(x)^{2}\leq\cosh(2x)$ for all $x\in\mathbb{R}$, we obtain

[TABLE]

We conclude the proof using the definition of $E_{2}$ and Parseval’s identity, similarly as what was achieved in (48). ∎

Remark 3.8**.**

One could derive a general formula for $C(d)$, however, we believe that this computation should be completed in the case to case scenario in order to obtain a sharper constant and to avoid unnecessary computations. Indeed, its computation relies on a combination of sums and integrals of exponentials that are straightforward to compute. We illustrate this analysis in the case of the Kawahara equation in Section 6. Moreover, once $C(d)$ is determined theoretically, then the upper bound given in (44) can easily be obtained numerically. Indeed, the Fourier series coefficients of $E_{1}$ and $E_{2}$ can be computed theoretically. Then, assuming that $V_{\alpha}$ only has a finite number of non-zero coefficients, then we only need a finite number of coefficients of $E_{1}$ and $E_{2}$ to compute $(V_{\alpha},V_{\alpha}*E_{i})$ ($i\in\{1,2\}$).

Remark 3.9**.**

Notice from (41) that $\mathcal{Z}_{u,2}^{\alpha}\approx\mathcal{Z}_{u,1}^{\alpha}$ if $d$ is sufficiently big and if $C(d)$ is computed accurately. Therefore, heuristically, if one chooses $d$ sufficiently big such that

[TABLE]

then using (31), one should obtain $\mathcal{Z}_{u}\ll 1$.

4 Existence of solutions via computer-assisted analysis

In this section, we present a computational method to prove constructively the existence of solutions of the PDE (3) in the space $H^{l}$. This approach begins with the construction of two important objects, namely an approximate solution $u_{0}\in H^{l}_{\Omega_{0}}$ and a linear operator $\mathbb{A}:L^{2}\to H^{l}$ which is an approximate inverse of $D\mathbb{F}(u_{0})$. Once these two objects are constructed, the (computer-assisted) proof of existence is obtained by showing that the Newton-like fixed point operator $\mathbb{T}:H^{l}\to H^{l}$ defined as $\mathbb{T}(u)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}u-\mathbb{A}D\mathbb{F}(u)$ possesses a unique fixed point in $\overline{B_{r}(u_{0})}$, where $B_{r}(u_{0})$ is the open ball of radius $r$ centered at $u_{0}$. This is achieved by verifying the hypotheses of a Newton-Kantorovich approach (see Section 4.2). In particular, the crucial hypothesis (59) to verify depends on three bounds $\mathcal{Y}_{0}$, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ that need to be computed explicitly. Using the construction of $\mathbb{A}$ presented in details in Section 3, we present in Section 4.3 the construction of such bounds using Fourier series objects. We begin this section by presenting the (computer-assisted) construction of the approximate solution $u_{0}\in H^{l}_{\Omega_{0}}.$

4.1 Construction of the approximate solution $u_{0}\in H^{l}_{\Omega_{0}}$

The construction of a numerical approximation requires considering finite dimensional projections. Let $N\in\mathbb{N}$, consider $X=\ell^{2}$ or $X=X^{l}$, and define the projections $\pi^{N}:X\to X$ and $\pi_{N}:X\to X$ as follows

[TABLE]

where $I^{N}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\{n\in\mathbb{Z}^{m},|n_{1}|\leq N,\dots,|n_{m}|\leq N\}.$ Then, assume that we numerically have access to an approximate solution to (19) given by $\tilde{U}_{0}\in X^{l}$ such that $\tilde{U}_{0}=\pi^{N_{0}}\tilde{U}_{0}$ (for some $N_{0}\in\mathbb{N}$). In practice, $\tilde{U}_{0}$ is obtained numerically by solving the equation $\pi^{N_{0}}F(\pi^{N_{0}}U)=0$, which is a finite dimensional projection of (19). Using (25), $\tilde{U}_{0}$ represents a function $\tilde{u}_{0}\in L^{2}_{\Omega_{0}}$ (that is a function in $L^{2}$ that vanishes outside ${\Omega_{0}}$). However, $\tilde{u}_{0}$ does not a priori have regularity and $\tilde{u}_{0}\notin H^{l}_{\Omega_{0}}$ in general. Consequently, one needs to build $u_{0}\in H^{l}_{\Omega_{0}}$, represented by a Fourier series $U_{0}\in X^{l}$, such that $U_{0}$ is close to $\tilde{U}_{0}$ in $X^{l}$. Moreover, $U_{0}$ needs to be represented on the computer, so we also require $U_{0}=\pi^{N_{0}}U_{0}$. Equivalently, we want to project $\tilde{U}_{0}$ into the set of vectors of size $N_{0}$ in $X^{l}$ representing functions in $H^{l}_{\Omega_{0}}$. This construction is achieved using some trace theory on hypercubes, that we introduce next.

4.1.1 Trace theorem for periodic functions

Denote by $\{\omega_{1}^{+},\dots,\omega_{m}^{+}\}\bigcup\{\omega_{1}^{-},\dots,\omega_{m}^{-}\}$ the $2m$ faces composing the surface of $\Omega_{0}$ defined in (17). In particular, we have

[TABLE]

The trace operator on $H^{s}(\Omega_{0})$ (for $s\in\mathbb{N}$) can be defined in the following theorem.

Theorem 4.1** **(Trace operator).

There exists an operator

[TABLE]

on $H^{s}(\Omega_{0})$, called the trace operator, which is continuous and satisfies

[TABLE]

for all $u\in C^{\infty}(\Omega_{0})$, where $\frac{\partial}{\partial n}$ is the normal derivative.

The idea is to first build $u_{\Omega_{0}}$, the restriction of $u_{0}$ on $\Omega_{0}$, such that $u_{\Omega_{0}}\in H^{l}_{per}(\Omega_{0})$ and such that $u_{\Omega_{0}}$ has a null trace of order $k$ (that is $\mathcal{T}_{k}(u_{\Omega_{0}})=0$). Then, if $k$ is big enough, by extending $u_{\Omega_{0}}$ by zero outside of $\Omega_{0}$, we obtain a smooth $u_{0}$ with support on $\Omega_{0}$. Lemma 4.4 below justifies such a reasoning. Note that $H^{k}_{0}(\Omega_{0})=Ker(\mathcal{T}_{k})$ denotes the set of trace-free elements in $H^{k}(\Omega_{0})$ (see [51] or [48] for a complete presentation). Consider the closed subspace $H^{k}_{0,per}(\Omega_{0})\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}H^{k}_{0}(\Omega_{0})\bigcap H^{k}_{per}(\Omega_{0})$ and denote by $X^{k}_{0}$ the correspondence of $H^{k}_{0,per}(\Omega_{0})$ in sequence space. Define $\tilde{X}^{k}$ as the equivalent of $X^{k}$ for sequences indexed on $\mathbb{Z}^{m-1}$

[TABLE]

and let

[TABLE]

Define a continuous trace operator $\Gamma_{k}:X^{k}\to\mathbb{X}^{k}$ by

[TABLE]

where

[TABLE]

for all $n=(n_{1},...,n_{i-1},n_{i+1},...,n_{m})\in\mathbb{Z}^{m-1}$. Note that $\prod_{j=0}^{k-1}\Gamma^{i,j}$ represents the trace component on the $i^{th}$ ($m-1$)-cube of $\partial\Omega_{0}$. Also, $\Gamma_{k}$ is the equivalent, in the restriction to periodic functions, to $\mathcal{T}_{k}$ defined in Theorem 4.1. In fact, the periodicity enables to simplify the definition of the trace operator and to only consider half the faces of $\Omega_{0}.$

For $m=1$, $\Gamma_{k}(X^{k})$ is finite-dimensional (it is a $k$-dimensional vector defined component-wise as the pointwise evaluations at $d$ of the $k$ first derivatives) so it is clearly closed. For $m>1$, the image of the trace operator is infinite dimensional and its closedness becomes less straightforward. Indeed, for $m=2$ and $m=3$, [58] gives a characterization of the image of the trace of polygons and polyhedrons and showed that the trace operator is closed, under some compatibility conditions. Hence for $m\leq 3$ $X^{k}_{0}(\Omega_{0})=R(\Gamma_{k}^{*})^{\perp}$ and the trace-theorem holds in the periodic case for $\Gamma^{*}_{k}$ defined with the compatibility conditions given in [58]. In particular, one can (in theory) build an orthonormal basis for $R(\Gamma^{*}_{k})$ to define an orthogonal projection on $X^{k}_{0}$. This construction is not trivial and would require to solve an infinite dimensional linear problem rigorously. In the general setting for $m>3$ we are not aware of any generalization of the previous results. From that perspective, we choose an alternative path and develop a finite dimensional trace theorem. We expose this result in the following section.

4.1.2 Finite dimensional trace theorem

Let $k\in\mathbb{N}$ and let $N\in\mathbb{N}$. Then let $U\in X^{k}$ such that $\pi^{N}(U)=U$ (so $U$ has a finite number of non-zero elements), we numerically want to build $U_{0}\in X^{k}_{0}$ such that $\|U-U_{0}\|_{X^{l}}$ is small. Moreover, we also require $\pi^{N}(U_{0})=U_{0}$ in order to obtain $U_{0}$ numerically, the end goal being to use interval arithmetics (see [55]).

First, define $\Gamma^{N}_{k}$ as

[TABLE]

for all $U\in X^{k}$. In fact, $\Gamma^{N}_{k}$ can be seen as a finite dimensional operator $\Gamma^{N}_{k}:\mathbb{C}^{(2N+1)^{m}}\to\mathbb{C}^{mk(2N+1)^{m-1}}$ that can be represented by an $mk(2N+1)^{m-1}$ by $(2N+1)^{m}$ matrix $\mathbb{\Gamma}$.

Now, we define $X^{N,k}_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}Ker(\Gamma^{N}_{k})$ and obtain that

[TABLE]

The above equality is an abuse of notation as elements in $R(\mathbb{\Gamma}^{*})$ are finite dimensional vectors. We need to pad with zeros these elements for the equality to be rigorous. In particular $X^{N,k}_{0}\subset X^{k}_{0}$. Now, a characterization of different projections on $X^{N}_{0}$, depending on some invertible matrices $D$, can be obtained. Again we abuse notation in the next Theorem 4.2 by considering elements $U\in X^{l}$ such that $\pi^{N}(U)=U$ as vectors in $\mathbb{C}^{(2N+1)^{m}}$.

Theorem 4.2**.**

Suppose $Rank(\mathbb{\Gamma}^{*})=r>0$ and $M$ is a $(2N+1)^{m}$ by $r$ matrix such that $R(\mathbb{\Gamma}^{*})=R(M)$, then if $D$ is an invertible matrix of size $(2N+1)^{m}$

[TABLE]

for all $U\in X^{l}$ such that $\pi^{N}(U)=U$.

Remark 4.3**.**

In the above Theorem 4.2, the choice of the matrix $D$ allows to choose the rate of decay of $U$. In particular, $P^{N}_{k}(U)$ minimizes $\|D^{-1}(U-V)\|_{2}$ with $V\in X^{k}$ such that $\pi^{N}(V)=V$. A good choice for $D$ is therefore to take a diagonal matrix with entries $\frac{1}{l(\tilde{n})}$ on the diagonal. Moreover, in practice we assume $2N+1>mk$. Indeed, as $\mathbb{\Gamma}$ is a $mk(2N+1)^{m-1}$ by $(2N+1)^{m}$ matrix, assuming $2N+1>mk$ allows the projection on $X^{N,k}_{0}$ to be non-trivial.

4.1.3 Construction of functions in $H^{l}_{\Omega_{0}}$

We go back to the initial goal of this section which was to build functions in $H^{l}_{\Omega_{0}}$ starting from a vector of Fourier series. Suppose that we have access to some approximation $\tilde{u}_{0}\in L^{2}$ for a solution given by

[TABLE]

for some $N_{0}\in\mathbb{N}$. Moreover, we denote $\tilde{U}_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(a_{n})_{n\in\mathbb{Z}}$ the Fourier series representation of $\tilde{u}_{0}$ (in particular $\pi^{N_{0}}(\tilde{U}_{0})=\tilde{U}_{0}$) and we want to build a smooth function $u_{0}\in H^{l}$ from $\tilde{u}_{0}\in L^{2}.$ This construction is achieved combining the next Lemma 4.4 and the construction (55).

Lemma 4.4**.**

Let $k\in\mathbb{N}$ and let $\widetilde{H}^{k}(\Omega_{0})\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\{u\in L^{2}(\Omega_{0}):E(u)\in H^{k}(\mathbb{R}^{m})\}$ where

[TABLE]

Then $\widetilde{H}^{k}(\Omega_{0})=H^{k}_{0}(\Omega_{0})$.

Proof.

The proof can be found in [48]. ∎

The previous Lemma 4.4 provides that a smooth extension by zero can be obtained for functions in $H^{k}_{0}(\Omega_{0})$. Consequently, if $k$ is chosen big enough so that $H^{l}\subset H^{k}(\mathbb{R}^{m})$, then the construction (55) allows to build $u_{0}\in H^{l}_{\Omega_{0}}$. Let us fix $k\in\mathbb{N}$ that satisfies the aforementioned requirement.

Now let $D$ be the matrix described in Remark 4.3, then let $U_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}P_{k}^{N_{0}}(\tilde{U}_{0})\in X^{N_{0},k}_{0}$ as in Theorem 4.2. The finite dimensional vector $U_{0}$ has a function representation $u_{0}$ in $H^{k}_{per,0}(\Omega_{0})$ and by Lemma 4.4 $u_{0}$ has a zero extension to a function in $H^{k}(\mathbb{R}^{m})\subset H^{l}$. To keep our notations, we still denote $u_{0}\in H^{k}(\mathbb{R}^{m})\subset H^{l}$ the zero extension on $\mathbb{R}^{m}$. By construction, we obtain $u_{0}\in H^{l}_{\Omega_{0}}$.

Remark 4.5**.**

Using some rigorous numerical interval arithmetics (such as IntervalArithmetics.jl [55] on Julia), Theorem 4.2 provides a way to rigorously obtain trace-free elements numerically. Therefore the approximate solution $u_{0}\in H^{l}_{\Omega_{0}}$ will be defined by its finite Fourier series representation using the construction (55).

4.2 Newton-Kantorovich approach

As explained in the previous Section 4.1.3, we now have a way to construct an approximate solution $u_{0}\in H^{l}_{\Omega_{0}}$ (via finite Fourier series in $X^{l}$) and thanks to Theorem 3.4, we can construct an approximate inverse $\mathbb{A}$ of $D\mathbb{F}(u_{0})$ using a Fourier series representation. We have now all the necessary ingredients to use a Newton-Kantorovich approach, which we present in a general setting, and which can either be applied to the PDE case described in Section 3 or to the constrained PDE case in Section 5.

Theorem 4.6** **(Newton-Kantorovich Theorem).

Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be two Banach spaces. Moreover, let $\mathbb{F}:X\to Y$ be a Fréchet differentiable map and let $x\in X$. Then, let $\mathbb{A}:Y\to X$ be an injective bounded linear operator and let $\mathcal{Y}_{0},\mathcal{Z}_{1},\mathcal{Z}_{2}$ be non-negative constants such that for all $r>0$

[TABLE]

If there exists $r>0$ such that

[TABLE]

then there exists a unique $\tilde{x}\in\overline{B_{r}(x)}\subset X$ such that $\mathbb{F}(\tilde{x})=0$, where $B_{r}(x)$ is the open ball of $X$ centered at $x$.

Proof.

The proof of the theorem can be found in [8]. In particular, the map $\mathbb{T}\colon\overline{B_{r}(x)}\to\overline{B_{r}(x)}$ defined by $T(u)=u-\mathbb{A}\mathbb{F}(u)$ is well-defined and contracting if (59) is satisfied. ∎

Remark 4.7**.**

Notice that if $r>0$ satisfies (59), then $(1-\mathcal{Z}_{1})r>\frac{1}{2}\mathcal{Z}_{2}(r)r^{2}+\mathcal{Y}_{0}>0$ and therefore $1-\mathcal{Z}_{1}>0$. Consequently, the assumption that $\mathbb{A}$ is injective becomes redundant in the set-up presented in Section 3.2 (see Theorem 3.4).

4.3 Computation of the bounds

In this section we expose the set-up to apply Theorem 4.6 in the case of a PDE given by (5). In other words, we choose $(X,\|\cdot\|_{X})=(H^{l},\|\cdot\|_{l})$ and $(Y,\|\cdot\|_{Y})=(L^{2},\|\cdot\|_{2})$ and we study $\mathbb{F}:H^{l}\to L^{2}$ defined in (5), which, under Assumptions 1 and 2, is smooth on $H^{l}$. We present explicit computations for the bounds $\mathcal{Y}_{0}$, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ using the constructions of $\mathbb{A}$ in Section 3 and of $u_{0}$ in Section 4.1. In particular, we derive upper bounds that only require the norm computation of finite dimensional objects. Such norms can then be obtained on a computer using the arithmetic on intervals (cf. [56] and [55]). Consequently, once $\mathbb{A}$ and $u_{0}\in H^{l}_{\Omega_{0}}$ are obtained, one can verify the hypotheses of Theorem 4.6 on a computer to prove the existence of solutions to (5).

Let us fix some $N,N_{0}\in\mathbb{N}$. $N_{0}$ represents the numerical size of the sequences (that is $U=\pi^{N_{0}}U$ for all the sequences $U$ we consider numerically) and $N$ represents the numerical size of the operators (that is $M=\pi^{N}M\pi^{N}$ for all the operators on Fourier series we consider numerically). Moreover, assume that $N_{0}>N$.

First, using the construction of Section 4.1 and (55), from a numerical approximation $\tilde{U}_{0}$ of $\pi^{N_{0}}F(\pi^{N_{0}}U)=0$, one can build an approximate solution $u_{0}\in H^{l}_{\Omega_{0}}$ represented by its Fourier series $U_{0}\in X^{l}$. Moreover, we have $U_{0}=\pi^{N_{0}}U_{0}$, that is $U_{0}$ only has a finite number of non-zero coefficients.

The second ingredient to apply Theorem 4.6, is the construction of $\mathbb{A}$. This is achieved using the construction of Section 3.1. First, we build $B_{\Omega_{0}}:\ell^{2}\to\ell^{2}$ numerically such that

[TABLE]

and such that $I_{d}+B_{\Omega_{0}}$ approximates the inverse of $I_{d}+DG(U_{0})L^{-1}$. Note that since (60) holds, $B_{\Omega_{0}}$ can be represented by a matrix. Then, using the construction of Section 3, we build $\mathbb{B}_{\Omega_{0}}=\mathcal{F}^{-1}(B_{\Omega_{0}})$ and an approximate inverse $\mathbb{A}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{L}^{-1}(I_{d}+\mathbb{B}_{\Omega_{0}}):L^{2}\to H^{l}$ of $D\mathbb{F}(u_{0}).$ Now that $u_{0}$ and $\mathbb{A}$ are fixed, and are both represented by finite dimensional objects, we present the computations for the bounds $\mathcal{Y}_{0}$, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ required to apply the Newton-Kantorovich Theorem 4.6. In particular, if $\mathcal{Z}_{1}<1$, then $\mathbb{A}$ has a bounded inverse (see Theorem 3.4).

We first take care of the bound $\mathcal{Z}_{1}$. This bound is decomposed into a first part $\mathcal{Z}_{u}$, that we compute thanks to Theorems 3.4 and 3.7, and a second part $Z_{1}$ that is based on Fourier series quantities. In particular, $Z_{1}$ can be computed numerically thanks to vectors and matrices norms. We summarize the obtained results in the following lemma.

Lemma 4.8** **(Bound $\mathcal{Z}_{1}$).

Let $Z^{N}_{1}$ and $Z_{1}$ be such that

[TABLE]

In particular, $\|I_{d}-ADF(U_{0})\|_{l}\leq Z_{1}$. Moreover, let $\mathcal{Z}_{u}$ satisfying (42). If $\mathcal{Z}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}Z_{1}+\mathcal{Z}_{u}$, then

[TABLE]

Proof.

Using (35), we have

[TABLE]

Then, using that $B=I_{d}+B_{\Omega_{0}}=\pi_{N}+\pi^{N}+B_{\Omega_{0}}$ and $B_{\Omega_{0}}=\pi^{N}B_{\Omega_{0}}\pi^{N}$, we get

[TABLE]

From (4), the monomials of $\mathbb{G}$ are of order at most $N_{\mathbb{G}}$, and hence $\pi^{N}DG(U_{0})=\pi^{N}DG(U_{0})\pi^{N_{\mathbb{G}}N}$. Moreover, as $L^{-1}$ is diagonal, we have $\pi^{N_{\mathbb{G}}N}L^{-1}=L^{-1}\pi^{N_{\mathbb{G}}N}$ and

[TABLE]

Then, we get

[TABLE]

Using again (4) and that $L^{-1}$ is diagonal, we have $DG(U_{0})L^{-1}\pi^{N}=\pi^{N_{\mathbb{G}}N}DG(U_{0})L^{-1}\pi^{N}$. This implies

[TABLE]

Finally, using (16) and recalling the notation (20), we have

[TABLE]

Therefore,

[TABLE]

However, using Young’s inequality for the convolution we have $\|\mathbb{V}_{\alpha}\|_{2}\leq\|V_{\alpha}\|_{1}.$ Moreover, using Assumption 2 and that $\partial^{\alpha}L^{-1}$ is diagonal, we have

[TABLE]

Therefore, using (61) and (62), we obtain that $\|I_{d}-ADF(U_{0})\|_{l}\leq Z_{1}$. The rest of the proof follows directly from Theorems 3.4 and 3.7. ∎

Remark 4.9**.**

Note that ${Z}_{1}$ (cf. (61) and (62)) is obtained via the computation of norms of vectors and matrices, which can be processed numerically. Moreover, since $U_{0}=\pi^{N_{0}}U_{0}$, we have $V_{\alpha}=\pi^{N_{\mathbb{G}}N_{0}}V_{\alpha}$ for each $\alpha\in J_{\mathbb{G}}$. Therefore,

[TABLE]

where $i\in\{1,2\}$. In particular, this implies that both $\mathcal{Z}_{u,1}^{\alpha}$ and $\mathcal{Z}_{u,2}^{\alpha}$ (satisfying (40) and (41)) can be obtained throughout a finite number of computations. Consequently, the bound $\mathcal{Z}_{1}$ can be fully computed throughout rigorous numerics (see [55] for instance).

Now, using the smoothness of the map $\mathbb{F}$, the bound $\mathcal{Z}_{2}$ satisfying (58) can directly be computed using the mean value theorem for Banach spaces, a result which we introduce next.

Lemma 4.10** **(Bound $\mathcal{Z}_{2}$).

Let $r>0$ and let $\mathcal{Z}_{2}(r)>0$ be such that

[TABLE]

where $\|\cdot\|_{\mathcal{B}(H^{l},L^{2})}$ is the operator norm from bounded linear operators in $H^{l}$ to bounded linear operators in $L^{2}.$ Then $\mathcal{Z}_{2}(r)r\geq\|\mathbb{A}\left(D\mathbb{F}(v)-D\mathbb{F}(u_{0}\right)\|_{l}$ for all $v\in\overline{B_{r}(u_{0})}\subset H^{l}.$

Proof.

Let $v\in B_{r}(u_{0})\subset H^{l}$, by definition of the $H^{l}$ norm, we have

[TABLE]

Then, using the mean value theorem for Banach spaces, there exists $h\in B_{r}(0)\subset H^{l}$ such that

[TABLE]

Finally using that $\|v-u_{0}\|_{l}\leq r$, we obtain

[TABLE]

Now, using (27), we have $\|\mathbb{B}\|_{2}=\max\{1,\|B\|_{2}\}$. Moreover, by construction we have

[TABLE]

Therefore, using the proof of Lemma 3.3, we get $\|B\|_{2}=\max\{1,\|\pi^{N}+B_{\Omega_{0}}\|_{2}\}$ as $B_{\Omega_{0}}=\pi^{N}B_{\Omega_{0}}\pi^{N}$ by assumption. This concludes the proof. ∎

In practice, the term $D^{2}\mathbb{F}(h)$ involves a mix of multiplication operators composed with differential operators. Therefore, its norm computation relies on estimates similar to the ones presented in Lemma 2.2. We illustrate such a computation in the case of the Kawahara equation in Lemma 6.3.

Finally, the computation for the bound $\mathcal{Y}_{0}$ is obtained using Parseval’s identity and the 2 norm of a vector. The next result provides the details for such a computation.

Lemma 4.11** **(Bound $\mathcal{Y}_{0}$).

Recall (19) and $N_{\mathbb{G}}\in\mathbb{N}$ from (4). Let $\mathcal{Y}_{0}>0$ be such that

[TABLE]

Then $\|\mathbb{A}\mathbb{F}(u_{0})\|_{l}\leq\mathcal{Y}_{0}.$

Proof.

Recall that we need to find $\mathcal{Y}_{0}>0$ such that

[TABLE]

Using (9), we have $\|u\|_{l}=\|\mathbb{L}u\|_{2}$ for all $u\in H^{l}$. This implies

[TABLE]

But now, because $u_{0}$ has support on $\Omega_{0}$, $\mathbb{F}(u_{0})$ also has a support on $\Omega_{0}$ and it has a Fourier series representation given by $F(U_{0})=LU_{0}+G(U_{0})-\Psi\in\ell^{2}$. Moreover, we have $\mathbb{B}=I_{d}+\mathbb{B}_{\Omega_{0}}$ and $\mathbb{B}_{\Omega_{0}}$ has a Fourier series representation given by $B_{\Omega_{0}}$ where $B_{\Omega_{0}}=\pi^{N}B_{\Omega_{0}}\pi^{N}.$ Therefore, as $\mathbb{F}(u_{0})$ has support on $\Omega_{0}$, using Parseval’s identity we obtain

[TABLE]

Moreover, we have that $U_{0}=\pi^{N_{0}}U_{0}$. In particular, $LU_{0}=\pi^{N}LU_{0}+(\pi^{N_{0}}-\pi^{N})LU_{0}$, $\Psi=\pi^{N}\Psi+\pi_{N}\Psi$ and $G(U_{0})=\pi^{N}G(U_{0})+\pi_{N}G(U_{0})$. We thus obtain

[TABLE]

Since $G$ is polynomial of order $N_{\mathbb{G}}$, $G(U_{0})=\pi^{N_{\mathbb{G}}N_{0}}G(U_{0})$ and so $\pi_{N}G(U_{0})=(\pi^{N_{\mathbb{G}}N_{0}}-\pi^{N})G(U_{0})$. This concludes the proof. ∎

Due to the foundation of our analysis on Fourier series, one can notice a lot of similarities with the computer-assisted proofs of periodic solutions using the Radii Polynomial approach (see [8]). More specifically, the bounds $\mathcal{Y}_{0}$, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ have corresponding bounds $Y_{0}$, $Z_{1}$ and $Z_{2}$ associated to the periodic problem on ${\Omega_{0}}.$ These similarities allow to derive a condition (65), under which a proof of a solution on $\mathbb{R}^{m}$ using Theorem 4.6 implies a proof of existence of a periodic solution.

Theorem 4.12** **(Existence of periodic solutions).

Let $\mathbb{A}:L^{2}\to H^{l}$ be defined in Section 3 and let $A:\ell^{2}\to X^{l}$ be its Fourier series correspondent. Let $\mathcal{Y}_{0},\mathcal{Z}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}Z_{1}+\mathcal{Z}_{u}$ (where $Z_{1}$ is defined in (62)) and $\mathcal{Z}_{2}$ be the associated bounds defined in Lemmas 4.11, 4.8 and 4.10 respectively. Assume that there exists $r>0$ such that (59) holds. Moreover, assume that

[TABLE]

Then, defining $Y_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\mathcal{Y}_{0}}{\sqrt{|{\Omega_{0}}|}}$ and $Z_{2}(r)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathcal{Z}_{2}(\sqrt{|{\Omega_{0}}|}r)\sqrt{|{\Omega_{0}}|}$, we have

[TABLE]

Letting $\tilde{r}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{r}{\sqrt{|{\Omega_{0}}|}}$, we get that

[TABLE]

and hence, there exists a unique solution $\tilde{U}$ to (19) in $\overline{B_{\tilde{r}}(U_{0})}$, where $B_{\tilde{r}}(U_{0})$ is the open ball in $X^{l}$ of radius $\tilde{r}$ and centered at $U_{0}.$

Proof.

First, using the proof of Lemma 4.11, we have

[TABLE]

This yields that $\|AF(U_{0})\|_{l}\leq Y_{0}$. Then $\|I_{d}-ADF(U_{0})\|_{l}\leq Z_{1}$ is true by definition of $Z_{1}$ in Lemma 4.8 and $\|A(DF(V)-DF(U_{0}))\|_{l}\leq Z_{2}(r)r$ for all $V\in B_{r}(U_{0})$ by assumption.

Now, suppose that $r>0$ satisfies (59) and let $\tilde{r}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{r}{\sqrt{|{\Omega_{0}}|}}$. Then let $T:X^{l}\to X^{l}$ be defined as

[TABLE]

We want to prove that $T:\overline{B_{\tilde{r}}(U_{0})}\to\overline{B_{\tilde{r}}(U_{0})}$ is well-defined and is a contraction. Notice that $Z_{1}\leq\mathcal{Z}_{1}$, then by construction of $Y_{0}$ and $Z_{2}$, we have

[TABLE]

using (59). Moreover, by construction of $\mathbb{A}$, we have that $A$ is injective if and only if $\mathbb{A}$ is injective, which is the case if (59) is satisfied (see Remark 4.7). Therefore, we apply Theorem 4.6 and obtain that $T:\overline{B_{\tilde{r}}(U_{0})}\to\overline{B_{\tilde{r}}(U_{0})}$ is well-defined and is a contraction. This concludes the proof. ∎

Once a solution $\tilde{u}\in B_{r}(u_{0})\subset H^{l}$ to (5) has been obtained through the use of Theorem 4.6, i.e bounds $\mathcal{Y}_{0},\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}(r)$ satisfying (59) for some $r>0$ have been obtained, then one can try to check if (65) is satisfied in order to obtain a solution to (19) in $X^{l}$. Using a Riemann sum argument (as illustrated in Remark 6.7), one can show that the condition (65) tends to the condition (58) as $d\to\infty$. Therefore, if $d$ is big enough, then (65) is usually easy to reach.

5 Constrained PDEs and Eigenvalue Problems

In this section, we introduce an extension of the analysis presented in Sections 3 and 4 to treat the case of a PDE coupled with an extra equation and for which we solve for an extra parameter. The most obvious example of this extension is the case of an eigenvalue problem (see Section 5.4) where one looks for an eigencouple $(\lambda,u)\in\mathbb{C}\times H^{l}$, where $u$ solves a linear PDE which depends on an eigenvalue $\lambda$ and where an extra equation is imposed to fix the scaling invariance of the eigenvector $u$. A second application of this extension is given by PDEs having conserved quantities leading to invariances, where one can use those conserved quantities as extra equations and use unfolding parameters to desingularize the problem (see [59, 60] for instance). The process of desingularization is useful to prove existence of solutions using a contraction mapping argument. A last example worth mentioning is the case of computing branches of solutions via a pseudo-arclength continuation algorithm, where a natural parameter of the PDE becomes an extra variable and where a hyperplane equation is appended to the problem (cf. [61, 62]).

While in this paper, we only consider the case of a PDE associated with one extra equation and one extra parameter, our approach can easily be generalized for multiple constraints with the same number of extra parameters. We begin by introducing the set-up of the problem.

5.1 Set-up of the augmented zero finding problem

Let $\phi\colon H^{l}\to\mathbb{C}$ be a bounded linear operator such that

[TABLE]

for all $u\in H^{l}$, where $\phi_{0}\in L^{2}_{\Omega_{0}}$ is fixed. We also denote by $\phi_{0}^{*}$ the dual of $\phi_{0}$, that is the bounded linear functional associated to $\phi_{0}$ in $L^{2}.$ In other words,

[TABLE]

We assume that the linear operator $\mathbb{L}=\mathbb{L}(\nu)$ and the differential operators in $\mathbb{G}=\mathbb{G}(\nu)$ (denoted $\mathbb{G}^{p}_{i,k}$ in Assumption 2) depend polynomially on a parameter $\nu\in\mathbb{C}$. As in Section 3, we fix some $u_{0}\in H^{l}$ such that supp$(u_{0})\subset\Omega_{0}$ ($u_{0}\in H^{l}_{\Omega_{0}}$). Moreover, we fix some $\nu_{0}\in\mathbb{C}$ such that $\mathbb{L}(\nu_{0})$ is invertible in order to ensure that Assumption 1 is still satisfied at $\nu_{0}$. In particular, we define the norm on $H^{l}$ as

[TABLE]

where $l_{\nu_{0}}$ ($|l_{\nu_{0}}|>0$) is the Fourier transform of $\mathbb{L}(\nu_{0})$. Denote by $H_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{C}\times H^{l}$ and $H_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{C}\times L^{2}$ the Hilbert spaces endowed respectively with the norms

[TABLE]

Denote $x=(\nu,u)\in H_{1}$ and define the augmented zero finding problem $\overline{\mathbb{F}}:H_{1}\to H_{2}$ as

[TABLE]

for all $x=(\nu,u)\in H_{1}$, where ${\mathbb{F}}(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{L}(\nu)u+\mathbb{G}(\nu,u)-\psi$ and for some fixed constant $\eta\in\mathbb{R}$. Denoting $x_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(\nu_{0},u_{0})$, we have that

[TABLE]

For the sake of simplicity of the presentation, denote

[TABLE]

The augmented zero finding problem (66) has a periodic boundary value problem correspondence. In fact, as $\phi_{0}\in L^{2}_{\Omega_{0}}$, we can define $\Phi\in\ell^{2}$ to be the corresponding Fourier series representation of $\sqrt{|\Omega_{0}|}\phi_{0}$ in $\ell^{2}$. Then we define $X_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{C}\times X^{l}$ and $X_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{C}\times\ell^{2}$ the corresponding Hilbert spaces associated to their natural norms

[TABLE]

Finally, we define a corresponding operator $\overline{F}:X_{1}\to X_{2}$ as

[TABLE]

where

[TABLE]

for all $(\nu,U)\in X_{1}$. The normalization by $\sqrt{|{\Omega_{0}}|}$ will make sense later in the proof of Lemma 5.2.

5.2 Approximate inverse of $D\overline{\mathbb{F}}(x_{0})$

In the same fashion as what is achieved in Section 3, we want to approximate the inverse of $D\overline{\mathbb{F}}(x_{0})$ by some operator $\overline{\mathbb{A}}:H_{2}\to H_{1}$ through the construction of an operator $\overline{\mathbb{B}}:H_{2}\to H_{2}$, where $\overline{\mathbb{A}}=\overline{\mathbb{L}}\overline{\mathbb{B}}.$ Moreover, we assume that $\overline{\mathbb{B}}=I_{d}+\overline{\mathbb{B}}_{\Omega_{0}}$ where

[TABLE]

In other words, we have

[TABLE]

where $b_{0}\in\mathbb{C}$, $b_{1},b_{2}\in L^{2}_{\Omega_{0}}$ and $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(L^{2})$. Using the notations of Section 3.1, denote by $\mathcal{B}_{\Omega_{0}}(H_{2})$ the subset of bounded linear operators on $H_{2}$ satisfying (68). One can derive the equivalent of Lemma 3.2 for operators in $\mathcal{B}_{\Omega_{0}}(H_{2})$.

Lemma 5.1**.**

Let $\overline{\mathbb{B}}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(H_{2})$ be a bounded linear operator such that

[TABLE]

where $b_{0}\in\mathbb{C}$, $b_{1},b_{2}\in L^{2}_{\Omega_{0}}$ and $\mathbb{B}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(L^{2})$. Then $\overline{\mathbb{B}}_{\Omega_{0}}$ has a unique representation as a Fourier series operator $\overline{B}_{\Omega_{0}}:X_{2}\to X_{2}$ given by

[TABLE]

where $B_{\Omega_{0}}$ is defined in Lemma 3.2 and $B_{1},B_{2}$ are the Fourier series representation of $\sqrt{|{\Omega_{0}}|}b_{1}$ and $\sqrt{|{\Omega_{0}}|}b_{2}$ respectively. Moreover, we abuse notation and for a given operator $\overline{B}_{\Omega_{0}}:X_{2}\to X_{2}$, we denote by

[TABLE]

its associated operator $\overline{\mathbb{B}}_{\Omega_{0}}:H_{2}\to H_{2}$ in $\mathcal{B}_{\Omega_{0}}(H_{2})$. Finally, we have

[TABLE]

Proof.

The construction of $\overline{B}_{\Omega_{0}}$ comes from the fact that Fourier series form a basis on $L^{2}({\Omega_{0}}).$ For the norm equality, let $x=(\nu,u)\in H_{2}$ and notice that

[TABLE]

Now using Parseval’s identity and denoting $U$ the Fourier series representation of $\mathbb{1}_{\Omega_{0}}u$, we have

[TABLE]

as $B_{1}$ and $B_{2}$ are the Fourier series representation of $\sqrt{|{\Omega_{0}}|}b_{1}$ and $\sqrt{|{\Omega_{0}}|}b_{2}$ respectively. Therefore,

[TABLE]

Finally, using Parseval’s identity, we obtain that

[TABLE]

We can similarly obtain that $\|\overline{\mathbb{B}}_{\Omega_{0}}\|_{H_{2}}\geq\|\overline{B}_{\Omega_{0}}\|_{X_{2}}$ and so $\|\overline{\mathbb{B}}_{\Omega_{0}}\|_{H_{2}}=\|\overline{B}_{\Omega_{0}}\|_{X_{2}}.$ ∎

The above construction allows to derive the equivalent of Lemma 3.5 in the case of constrained PDEs. More specifically, the next Lemma 5.2 provides control on the approximation $\overline{\mathbb{A}}$ for the inverse of $D\overline{\mathbb{F}}(x_{0})$.

Lemma 5.2**.**

Let $\overline{\mathbb{A}}:H_{2}\to H_{1}$ be a bounded linear operator such that $\overline{\mathbb{A}}=\overline{\mathbb{L}}^{-1}\overline{\mathbb{B}}$ where $\overline{\mathbb{B}}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}I_{d}+\overline{\mathbb{B}}_{\Omega_{0}}:H_{2}\to H_{2}$. Moreover, assume that $\overline{\mathbb{B}}_{\Omega_{0}}\in\mathcal{B}_{\Omega_{0}}(H_{2})$ and let $\overline{B}_{\Omega_{0}}:X_{2}\to X_{2}$ be its Fourier series representation given in Lemma 5.1. Finally, denote $\overline{B}=I_{d}+\overline{B}_{\Omega_{0}}:X_{2}\to X_{2}$ and $\overline{A}=\begin{pmatrix}1&0\\ 0&L^{-1}\end{pmatrix}\overline{B}:X_{2}\to X_{1}.$

Let $Z_{1}>0$ and $\mathcal{Z}_{u}>0$ be such that

[TABLE]

If $\mathcal{Z}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}Z_{1}+\mathcal{Z}_{u}$, then

[TABLE]

Moreover, if $\mathcal{Z}_{1}<1$, then both $\overline{\mathbb{A}}:H_{2}\to H_{1}$ and $D\overline{\mathbb{F}}(x_{0}):H_{1}\to H_{2}$ have a bounded inverse and

[TABLE]

Proof.

First, notice that

[TABLE]

as $\|u\|_{H_{1}}=\|\overline{\mathbb{L}}u\|_{H_{2}}$ for all $u\in H_{1}$. Then using that $\overline{\mathbb{L}}\overline{\mathbb{A}}=\overline{\mathbb{B}}$, we get

[TABLE]

by definition of $\phi.$ Then, using (16), we obtain

[TABLE]

Recall that both $\phi_{0}$ and $\partial_{\nu}\mathbb{F}(x_{0})$ are in $L^{2}_{\Omega_{0}}$. Also, recall that $\sqrt{|{\Omega_{0}}|}\phi_{0}$ has a Fourier series representation $\Phi\in\ell^{2}$. Similarly, $\partial_{\nu}\mathbb{F}(x_{0})$ also has a Fourier series representation given by $\partial_{\nu}F(\nu_{0},U_{0})$. Equivalently, $\sqrt{|{\Omega_{0}}|}\partial_{\nu}\mathbb{F}(x_{0})$ has a Fourier series representation given by $\sqrt{|{\Omega_{0}}|}\partial_{\nu}F(\nu_{0},U_{0})=\partial_{\nu}\tilde{F}(\frac{\nu_{0}}{\sqrt{|{\Omega_{0}}|}},U_{0})$, where $\tilde{F}$ is defined in (67).

Now, let $x=(\nu,u)\in H_{2}$ and denote $U\in\ell^{2}$ the Fourier series representation of $\mathbb{1}_{\Omega_{0}}u$. Then,

[TABLE]

using the reasoning of (70).

Therefore, using Parseval’s identity and the analysis of Theorem 3.4, we have

[TABLE]

Recall that $\partial_{u}\mathbb{G}(x_{0})\mathbb{L}^{-1}$ is compact on $L^{2}$ (see Lemma 3.1). Moreover, $\begin{pmatrix}-1&\phi(\mathbb{L}^{-1}\cdot)\\ \partial_{\nu}\mathbb{F}(x_{0})&\partial_{u}\mathbb{G}(x_{0})\mathbb{L}^{-1}\end{pmatrix}:H_{2}\to H_{2}$ is also a compact as $\nu\to\nu\partial_{\nu}\mathbb{F}(x_{0})$ and $u\to\phi(\mathbb{L}^{-1}u)$ have a finite dimensional domain and a finite dimensional co-domain respectively. Therefore,

[TABLE]

is a Fredholm operator. The rest of the proof follows identically as the proof of Theorem 3.4. ∎

The previous Lemma 5.2 provides a way to compute the defect $\|I_{d}-\overline{\mathbb{A}}D\overline{\mathbb{F}}(x_{0})\|_{l}$ under the construction $\overline{\mathbb{A}}=\overline{\mathbb{L}}^{-1}\overline{\mathbb{B}}$ and $\overline{\mathbb{B}}$ is determined by (68). Moreover, the previous Lemma 5.2 provides that $\overline{\mathbb{A}}$ is injective if $\mathcal{Z}_{1}<1$ (which is required in order to use Theorem 4.6). Therefore, using Remark 4.7, if (59) is satisfied then assuming that $\overline{\mathbb{A}}$ is injective becomes redundant. In practice, one needs to build $\overline{B}_{\Omega_{0}}$ in such a way that $\overline{A}=\begin{pmatrix}1&0\\ 0&L^{-1}\end{pmatrix}(I_{d}+\overline{B}_{\Omega_{0}})$ approximates the inverse of $D\overline{F}(\frac{\nu_{0}}{\sqrt{|{\Omega_{0}}|}},U_{0})$. Then we can build $\overline{\mathbb{B}}_{\Omega_{0}}=\mathcal{F}^{-1}(\overline{B}_{\Omega_{0}})$, which leads to the construction of $\overline{\mathbb{A}}=\overline{\mathbb{L}}^{-1}(I_{d}+\overline{\mathbb{B}}_{\Omega_{0}})$.

Remark 5.3**.**

Similarly as Remark 3.5, if one has access to the quantity $\|D\overline{F}(\frac{\nu_{0}}{\sqrt{|{\Omega_{0}}|}},U_{0})^{-1}\|_{X_{2},X_{1}}$, then

[TABLE]

where

[TABLE]

5.3 Bounds for PDEs with constraints

The analysis presented in Section 4.3 can naturally be extended for the constrained PDE case. We present in this section the details for the computations of the bounds in Theorem 4.6 applied to (66).

First, define $\phi_{0}\in L^{2}$ as follows

[TABLE]

By construction we have that $\phi_{0}\in L^{2}$ and supp$(\phi_{0})\subset{\Omega_{0}}$, which are the prerequisites introduced in Section 5. Denote by $\Phi\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(z_{n})_{n\in\mathbb{Z}^{m}}$ the Fourier series representation of $\phi_{0}$. Moreover, we consider that $x_{0}=(\nu_{0},u_{0})\in H_{1}$ is given and $u_{0}\in H^{l}_{\Omega_{0}}$ is constructed using Section 4.1.3.

We consider the operator $\overline{\mathbb{F}}$ defined in Section 5.4 as well as its corresponding operator on Fourier series

[TABLE]

where $\Psi$ is the Fourier series representation of $\mathbb{1}_{\Omega_{0}}\psi$ defined earlier. As in Section 5, we denote ${L}(\nu_{0})$ as $L$ to simplify notations. Moreover, assume that $\overline{\mathbb{A}}:H_{2}\to H_{1}$ is fixed and was constructed using the approach of Section 5.2.

In a similar fashion as Section 4.3, we can compute the bound $\mathcal{Y}_{0}$, $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ in the case of a PDE with constraints. The computation of $\mathcal{Z}_{1}$ has already been detailed in Lemma 5.2. Moreover, the computation of $\mathcal{Z}_{2}$, similarly as what was achieved in Lemma 4.10, can be obtained using the mean value theorem for Banach spaces. The main difference lies in the $\mathcal{Y}_{0}$ bound. We expose the details of its computation in the following lemma.

Lemma 5.4**.**

Let $\mathcal{Y}_{0}>0$ be such that

[TABLE]

where $M:X_{2}\to X_{2}$ is defined as $M\begin{pmatrix}\nu\\ U\end{pmatrix}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\begin{pmatrix}\nu\\ \sqrt{|{\Omega_{0}}|}U\end{pmatrix}$. Then $\|\overline{\mathbb{A}}\overline{\mathbb{F}}(x_{0})\|_{H_{1}}\leq\mathcal{Y}_{0}.$

Proof.

The proof follows the same steps as the proof of Lemma 4.11. By definition of the norm on $H_{1}$ and using (2), we have

[TABLE]

Now, recall that $\overline{\mathbb{B}}=I_{d}+\overline{\mathbb{B}}_{\Omega_{0}}$ and using that $\overline{\mathbb{F}}(x_{0})=\begin{pmatrix}1&0\\ 0&\mathbb{1}_{\Omega_{0}}\end{pmatrix}\overline{\mathbb{F}}(x_{0})$, we obtain that

[TABLE]

Then, notice that $\overline{\mathbb{B}}\begin{pmatrix}1&0\\ 0&\mathbb{1}_{\Omega_{0}}\end{pmatrix}$ as a Fourier series representation given by $I_{d}+\overline{B}_{\Omega_{0}}$ using Lemma 5.1. Therefore, we can use (70) and we get

[TABLE]

Remark 5.5**.**

Similarly as what was achieved in Theorem 4.12, the existence of a zero of $\overline{\mathbb{F}}$ in $H_{2}$ using Theorem 4.6 implies the existence of a zero of $\overline{F}$ under a mild assumption of the bound $\mathcal{Z}_{2}$.

5.4 Application to the proof of isolated eigenvalues of the Jacobian

As described earlier, one of the main application of the present section is eigenvalue problems. In this section, we define the needed operators for the proof of eigenvalues of $\mathbb{L}+D\mathbb{G}(\tilde{u})$, where $\tilde{u}\in H^{l}$. In particular, supposing that $\mathbb{F}(\tilde{u})=0$ and that one is interested in the dynamics of $\partial_{t}v=\mathbb{F}(v)$ around $\tilde{u}$, the eigenvalues of $\mathbb{L}+D\mathbb{G}(\tilde{u})$ will provide the stability of the stationary solution $\tilde{u}$. In particular, if one eigenvalue of $\mathbb{L}+D\mathbb{G}(\tilde{u})$ has a positive real part, then, under some possible additional conditions, one obtain that $\tilde{u}$ is unstable.

Let $v_{0}\in H^{l}_{{\Omega_{0}}}$ be an approximation of an eigenvector, let $\nu_{0}\in\mathbb{C}$ be its associated approximate eigenvalue and denote $x_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(\nu_{0},v_{0})$. Let us denote $\phi_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{L}v_{0}\in L^{2}_{\Omega_{0}}$ and define $\phi:H^{l}\to\mathbb{C}$ as

[TABLE]

for all $u\in H^{l}$. Then we define the following zero finding problem $\overline{\mathbb{F}}:H_{1}=\mathbb{C}\times H^{l}\to H_{2}=\mathbb{C}\times L^{2}$ as

[TABLE]

for all $x=(\nu,v)\in H_{1}$. Therefore by defining $\mathbb{L}(\nu)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathbb{L}-\nu I_{d}$, we need to make sure that $\mathbb{L}-\nu_{0}I_{d}$ is invertible to enter the set up of the previous section. This criterion will provide the range of eigencouples we can hope to prove.

If the support of $\tilde{u}$ is not a subset of $\Omega_{0}$, we cannot apply directly the analysis of Section 5.2 to $D\overline{\mathbb{F}}(x_{0})$. However, suppose that $\tilde{u}$ was obtained by applying Theorem 4.6 to (5) with some $u_{0}\in H^{l}_{\Omega_{0}}$. In practice, $\|\tilde{u}-u_{0}\|_{l}$ is supposedly small. Then we can define $\overline{\mathbb{F}}_{0}$ as

[TABLE]

Now we can apply Lemma 5.2 and build an approximate inverse $\overline{\mathbb{A}}$ for $D\overline{\mathbb{F}}_{0}(x_{0})$. In particular, Lemma 5.2 allows to compute an upper bound for $\|I_{d}-\overline{\mathbb{A}}D\overline{\mathbb{F}}_{0}(x_{0})\|_{H_{1}}$. Moreover, the following lemma allows to pass from $\overline{\mathbb{F}}$ to $\overline{\mathbb{F}}_{0}$ and obtain the needed bounds to apply Theorem 4.6 to $\eqref{eq : eigenvalue problem}$.

Lemma 5.6**.**

Let $\overline{\mathbb{A}}:H_{2}\to H_{1}$, then we have that

[TABLE]

Moreover,

[TABLE]

Proof.

First, notice that

[TABLE]

Then, the first inequality is obtained using the triangle inequality. Now, we have

[TABLE]

However, by definition of $\overline{\mathbb{F}}_{0}$, we have

[TABLE]

Therefore, we obtain

[TABLE]

Notice that, using Lemma 5.6 combined with Lemma 5.2 and Lemma 5.4, one can compute the bounds of Theorem 4.6 applied to $\eqref{eq : eigenvalue problem}$. We illustrate these computations in Section 6.3 for the proof of eigencouples of the linearization of the Kawahara equation about a proven soliton. In particular, the proof of the eigencouples provides the necessary ingredients for the proof of stability of the soliton (cf. Proposition 6.14).

6 Application to Kawahara equation

We apply our techniques to a one-dimensional example: the Kawahara equation. This model can be seen as a standard third order KdV equation complexified with an extra fifth order term taking into account the surface tension. This is a model for the study of fluid dynamics which simplifies the Whitham equation. In particular, the coefficients $a(T)$ and $b(T)$ below come from the Taylor expansion of the full dispersion relation in the Whitham equation. Details on the study of such equations can be found in [63].

In fact, the Kawahara equation is known to have soliton solutions (traveling wave with constant velocity of propagation and vanishing at infinity). In particular, soliton solutions have been proven to exist in [64] and [65]. The proof is theoretical and is obtained via a constrained minimization problem. The solutions are ground states and are proven to be exponentially decaying to zero. Moreover, some soliton in $\text{sech}^{4}(x)$ are known explicitly (see [66] and [67]). These solutions are obtained using a tanh-type of method and using auxiliary equations. Finally, numerical experiments have also been conducted on the study of soliton solutions (see [68] or [69]) in order to understand their formation and dynamics. Our results provide constructed proofs of solutions (based on a numerical approximation $u_{0}$) of the Kawahara equation. In this sense, we provide new results for the Kawahara equation and our proofs allow to make the bridge between some known theoretical and numerical results.

We introduce the Bond number $T\geq 0$ and look for solitons in

[TABLE]

where $a(T)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{1}{6}(1-3T)$ and $b(T)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{1}{360}(19-30T-45T^{2})$. By using the standard traveling wave change of variable $x=y-ct$ and integrating once (the constants of integration disappear as we look for solutions going to zero at infinity), we study the following ordinary differential equation

[TABLE]

Normalizing the linear part, we define $\lambda_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}-\frac{a(T)}{1-c}$, $\lambda_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{b(T)}{1-c}$ and $\lambda_{3}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{3}{4(1-c)}$. Finally, we obtain the following zero finding problem

[TABLE]

where $\mathbb{F}(u)=\mathbb{L}u+\mathbb{G}(u)$ and

[TABLE]

At this point, we verify the required Assumptions 1 and 2. First, denote by $l:\mathbb{R}\to\mathbb{R}$ the Fourier transform of $\mathbb{L}$ and $l(\xi)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}1+\lambda_{1}(2\pi\xi)^{2}+\lambda_{2}(2\pi\xi)^{4}$ for all $\xi\in\mathbb{R}$. In particular, the linear part $\mathbb{L}$ is invertible if and only if $(1-c)-a(T)\xi^{2}+b(T)\xi^{4}>0$ for all $\xi\in\mathbb{R}$. After computing the roots of the previous fourth order polynomial, if

[TABLE]

then Assumption 1 is satisfied and we can define the Hilbert space $H^{l}$ given in (8). Consequently, we suppose in the rest of this paper that $c$ and $T$ are chosen such that (76) is satisfied. In particular, notice that $\lambda_{1},\lambda_{2},\lambda_{3}>0$ and $l(\xi)\geq 1$ for all $\xi\in\mathbb{R}$. Then, we have $\mathbb{G}(u)=\lambda_{3}I_{d}(u)I_{d}(u)=\mathbb{G}^{1}_{2}(u)\mathbb{G}^{2}_{2}(u)$ where $\mathbb{G}^{1}_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\lambda_{3}I_{d}$ and $\mathbb{G}^{2}_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}I_{d}$. Using the notations of Assumption 2, we have $g^{1}_{2}=\frac{\lambda_{3}}{l}$ and $g^{2}_{2}=\frac{1}{l}$. Consequently, Assumption 2 is satisfied if and only if $\frac{1}{l}\in L^{1}$, which is trivial to verify. Both Assumptions 1 and 2 are satisfied and the analysis derived in the previous Sections is applicable to (75).

Note that the set of solutions in $H^{l}$ of equation (75) possesses a natural translation invariance. One way to fix this invariance is to restrict to even solutions. Moreover, we look for real valued solutions as the Kawahara equation is real-valued. Therefore we define $H^{l}_{e}$ as the subspace of even functions in $H^{l}$

[TABLE]

and look for zeros of $\mathbb{F}$ in $H^{l}_{e}.$ Similarly, $L^{2}_{e}$ is the subspace of even functions in $L^{2}$. We also define the restriction of our operators on the even restriction. We denote $\mathbb{L}_{e}:H^{l}_{e}\to L^{2}_{e}$ the restriction of $\mathbb{L}$, $\mathbb{F}_{e},\mathbb{G}_{e}:H^{l}_{e}\to L^{2}_{e}$ the ones of $\mathbb{F}$ and $\mathbb{G}$ respectively. We define in the same manner the even restrictions for the operators and spaces on Fourier series. Equation (75) is turned into

[TABLE]

Note that the analysis derived in the previous sections is still applicable.

Remark 6.1**.**

There exists values for $c$ and $T$, other than the ones given in (76), for which $l(\xi)>0$ for all $\xi\in\mathbb{R}.$ For these values, the analysis derived in Section 6 is still applicable under some mild modifications.

6.1 Computation of the theoretical bounds

Denote by $\Omega_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}(-d,d)$ our domain of study and as in Section 4.3 denote by $U_{0}$ our numerical guess that was projected on $X^{N,4}_{0}\bigcap X^{l}_{e}$ and $u_{0}\in H^{l}_{e}$ its function representation on $\mathbb{R}$. In particular, supp$(u_{0})\subset\Omega_{0}$ by construction. We present in the following subsections the explicit computations of the bounds $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$ using the analysis derived in Sections 3 and 4. Note that the computation of the bound $\mathcal{Y}_{0}$ is already explicited in Lemma 4.11.

6.1.1 Computation of $\mathcal{Z}_{2}$

We begin with the computation of the bound $\mathcal{Z}_{2}$. First, notice that the proof of Lemma 2.2 implies that for $\kappa\geq\max_{\xi\in\mathbb{R}}\frac{\lambda_{3}}{l(\xi)}\|\frac{1}{l}\|_{2}$ then

[TABLE]

for all $w,v\in H^{l}$. In order to obtain an explicit value for $\kappa$, we use the following Proposition 6.2.

Proposition 6.2**.**

Under (76), we have

[TABLE]

Proof.

The computation for $\displaystyle\max_{\xi\in\mathbb{R}}{\frac{1}{l(\xi)}}$ is a simple study of the extrema of the polynomial $l$. For the value of $\|\frac{1}{l}\|_{2}$ we notice that

[TABLE]

In particular, one can compute the following

[TABLE]

Using Proposition 6.2, we fix some $\kappa>0$ such that

[TABLE]

and we have $\|\lambda_{3}wv\|_{2}\leq\kappa\|w\|_{l}\|v\|_{l}$ for all $w,v\in H^{l}$. Moreover, we can compute an upper bound for $\mathcal{Z}_{2}$ defined in Theorem 4.6.

Lemma 6.3**.**

Let $r>0$ and let $\mathcal{Z}_{2}(r)$ be defined as

[TABLE]

where $\kappa$ satisfies (79). Then, $\|\mathbb{A}\left(D\mathbb{F}_{e}(u_{0})-D\mathbb{F}_{e}(v)\right)\|_{l}\leq\mathcal{Z}_{2}(r)r$ for all $v\in\overline{B_{r}(u_{0})}\subset H^{l}_{e}$.

Proof.

Using Lemma 4.10, we can choose

[TABLE]

However, we have $D^{2}\mathbb{F}_{e}(h)w=2\lambda_{3}\mathbb{w}$ for all $w\in H^{l}_{e}$ and all $h\in B_{r}(u_{0}).$ Moreover, using (78), we have $\|\lambda_{3}wv\|_{2}\leq\kappa\|w\|_{l}\|v\|_{l}$ for all $v\in H^{l}_{e}$. Consequently, we get

[TABLE]

6.1.2 Computation of $\mathcal{Z}_{1}$

Using Lemma 4.8, one can explicitly compute a value of $Z_{1}$. To obtain a value for $\mathcal{Z}_{1}$, it remains to compute an upper bound $\mathcal{Z}_{u}$. This is achieved using Theorem 3.7. First, using the definition of $\mathbb{G}$ in the Kawahara equation, we have

[TABLE]

Denote by

[TABLE]

the Fourier series representation of $v_{0}$ as defined in (25). Now, Theorem 3.7 requires the computations of $a>0$ and of the functions $(f_{\alpha})$ defined in Lemma 3.6. However, as $\mathbb{G}$ does not possess derivatives, we only need to compute $f_{0}(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathcal{F}^{-1}(\frac{1}{l(\xi)})$.

Now, straightforward computations show that under (76), the roots of $l$ are given by

[TABLE]

for some $a>0$ and some $b\geq 0$. Note that $a\neq 0$ since $l>0$ by assumption. The next lemma then provides the computation of $f_{0}$ using some basic Fourier analysis.

Lemma 6.4**.**

Let $f_{0}(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathcal{F}^{-1}(\frac{1}{l(\xi)})(x)$ and suppose that $l$ has 4 roots $\{b\pm ia,-b\pm ia\}$ with $a>0$ and $b\geq 0$. Then we have

[TABLE]

In particular,

[TABLE]

for all $x\in\mathbb{R}$ where $C_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{a+b}{4|\lambda_{2}|ab(a^{2}+b^{2})}.$

Proof.

First, notice that if $f\in L^{2}$, then

[TABLE]

for all $\xi\in\mathbb{R}$ where $\tilde{f}(x)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}f(2\pi x)$ for all $x\in\mathbb{R}.$ Then, we have

[TABLE]

Using [70] (Section 2.3.14) combined with (81), we obtain the expression for the above Fourier inverse

[TABLE]

Now noticing that

[TABLE]

we conclude the proof. ∎

Now that the constants $a$ and $C_{0}$ are given in the previous Lemma 6.4, the bound $\mathcal{Z}_{u}$ can be computed using Theorem 3.7.

Lemma 6.5**.**

Let $C_{0}$ and $a$ be defined in Lemma 6.4. Moreover, let $E$ be the Fourier series representation of $\cosh(2ax)$ on ${\Omega_{0}}$ and define

[TABLE]

Then

[TABLE]

where

[TABLE]

Moreover, if $\mathcal{Z}_{u}\geq\max\{1,\|B\|_{1}\}\left((\mathcal{Z}_{u,1})^{2}+(\mathcal{Z}_{u,2})^{2}\right)^{\frac{1}{2}}$, then $\mathcal{Z}_{u}$ satisfies $\|(\mathbb{L}^{-1}-\mathbb{K}_{0})\mathbb{v}_{0}\mathbb{B}^{*}\|_{2}\leq\mathcal{Z}_{u}$, where $\mathbb{K}_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\mathcal{F}^{-1}(L^{-1})$ using the notations of Lemma 3.2.

Proof.

First, notice that since $l$ is real valued, then $\mathbb{L}^{*}=\mathbb{L}$ and $L^{*}=L$. Similarly, we have $v_{0}=v_{0}^{*}$ as $u_{0}$ is real-valued. Therefore, $\|((\mathbb{L}^{-1})^{*}-\mathbb{K}_{0})\mathbb{v}_{0}^{*}\|_{2}=\|(\mathbb{L}^{-1}-\mathbb{K}_{0})\mathbb{v}_{0}\|_{2}$. Let $u\in L^{2}_{e}$ such that $\|u\|_{2}=1$. Then, define $v\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}v_{0}u$. By construction we have $v\in L^{2}_{e}$ and supp$(v)\subset{\Omega_{0}}.$ Using Theorem 3.7 and more specifically equation (52), we need to compute

[TABLE]

where

[TABLE]

and where we used that $v\in L^{2}_{e}$. Notice that

[TABLE]

where the last equality follows from Lemma 8.1 in the Appendix.

Now, notice that since $v\in L^{2}_{e}$, we have

[TABLE]

Then, using Cauchy-Schwartz inequality, we get

[TABLE]

Moreover, using that $v_{0}\in L^{2}_{e}$, we have

[TABLE]

where we used Parseval’s identity. Then, notice that

[TABLE]

for all $n\in\mathbb{N}$. This implies that

[TABLE]

for all $n\in\mathbb{N}$. Using (85), we obtain

[TABLE]

Moreover, we have

[TABLE]

Therefore, combining (6.1.2), (6.1.2), (84), (6.1.2) and (87) we get

[TABLE]

where $C(d)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}4d+\frac{4e^{-ad}}{a(1-e^{-\frac{3ad}{2}})}+\frac{2}{a(1-e^{-2ad})}$. We conclude the proof using Theorem 3.7 and Lemma 4.8. ∎

6.2 Numerical results

Let $T=0.35$ and $c=0.9$ (in particular $\frac{1}{3}<T<\frac{-10+\sqrt{480}}{30}$ and $c<1-\frac{a(T)^{2}}{4b(T)}$) and define $w_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{-\lambda_{1}}{2\lambda_{2}}$ and $w_{2}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\sqrt{|\lambda_{1}^{2}-4\lambda_{2}|}}{4\lambda_{2}}$. Then, denote

[TABLE]

It is well know that the classical KdV equation

[TABLE]

has even soliton solutions in $\text{sech}^{2}(x)$. Using these solutions, we can initialize Newton’s method and then converge to some approximate solutions of the Kawahara equation. That enables us to build a vector of Fourier series $\tilde{U}_{0}$. Then we build the trace operator $\Gamma_{4}$ (which is a matrix) as defined in (54). In particular, we have that

[TABLE]

for all $U\in X^{l}$ such that $U=\pi^{N}(U)$ and all $j\in\{0,1,2,3\}$. Using Theorem 4.2 and (55), we can project $\tilde{U}_{0}$ in the kernel of $\Gamma_{4}$ and obtain $U_{0}\in X^{N,4}_{0}.$ In particular, using the notations of the theorem, we use $M=(\Gamma_{4})^{*}$ and $D=\pi^{N}L^{-1}\pi^{N}$. The computations in (55) are achieved using IntervalArithmetic on Julia [55] and the code is available at [47]. In particular, $U_{0}$ has a representation $u_{0}\in H^{l}_{e}$ such that supp$(u_{0})\subset\Omega_{0}.$

Using the approach of Section 4.3, we were able to obtain the following bounds

[TABLE]

Moreover, using Theorem 4.6, we prove that there exists a unique solution in some ball of $H^{l}_{e}$ around $u_{0}$. The representation of $u_{0}$ is given in Figure 1 below.

Theorem 6.6**.**

(Proof of a soliton in the Kawahara equation) Let $r_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}2.27\times 10^{-14}$, then there exists a unique solution $\tilde{u}$ to (77) in $\overline{B_{r_{0}}(u_{0})}\subset H^{l}_{e}$ and we have that $\|\tilde{u}-u_{0}\|_{l}\leq r_{0}$. Moreover, the solution is unique in $\overline{B_{0.015}(u_{0})}$. Finally, there exists a unique $\tilde{U}\in\overline{B_{\frac{r_{0}}{\sqrt{|\Omega_{0}|}}}(U_{0})}\subset X^{l}_{e}$ such that $\tilde{U}$ is a solution to the periodic equivalent of (77) on $\Omega_{0}$.

Proof.

The rigorous computations for the needed bounds of Theorem 4.6 are obtained via [47]. In particular, the proof of $\tilde{u}$ is a direct application of the aforementioned theorem. For the proof of $\tilde{U}\in X^{l}_{e}$, we use Theorem 4.12. Let $r>0$ and let $V\in\overline{B_{r}(U_{0})}$, then for all $W\in X^{l}_{e}$ we have

[TABLE]

But similarly as the reasoning of Lemma 2.2, we obtain that

[TABLE]

Numerically we made sure to build $\kappa$ such that

[TABLE]

and therefore

[TABLE]

Now, choosing $\mathcal{Z}_{2}(r)\geq 2\kappa\max\{1,\|\pi^{N}+B_{\Omega_{0}}\|_{2}\}$, we conclude the proof combining Lemma 6.3 and Theorem 4.12. ∎

Remark 6.7**.**

Notice that the condition (65) in Theorem 4.12 is transformed into (88) in the previous lemma. However, using a Riemann sum argument, notice that

[TABLE]

as $d\to\infty$. Consequently, as $d$ increases, choosing $\kappa$ satisfying (88) is less and less constraining (compared to $\kappa\leq\|\frac{1}{l}\|_{2}$ in Lemma 2.2).

6.3 First three eigencouples of $\mathbb{L}+D\mathbb{G}(\tilde{u})$

In this section, we return to the original setting where $\mathbb{L}$ and $D\mathbb{G}(\tilde{u})$ are defined on the full $H^{l}$ without any restriction to even functions. Here, $\tilde{u}$ denotes the soliton obtained in Theorem 6.6. We first notice that $\mathbb{L}+D\mathbb{G}(\tilde{u})$ only has real eigenvalues by self-adjointness on $L^{2}$. Therefore we replace $\mathbb{C}$ by $\mathbb{R}$ when using the results of Sections 5.4 and 5.3.

First, we compute (see [47]) numerical approximations of the first three eigencouples of $L+DG(U_{0})$ which we denote by $(\nu_{1},U_{1}),(\nu_{2},U_{2}),(\nu_{3},U_{3})$ with $\nu_{1}\leq\nu_{2}\leq\nu_{3}$. Then, for all $1\leq i\leq 3$, we project $U_{i}$ into $X^{N,k}_{0}$ and still call $U_{i}$ the projection. In particular, $U_{i}$ has a representation $u_{i}\in H^{l}$ such that supp$(u_{i})\subset\Omega_{0}$.

We want to use the set up of Section 5.4 to prove that the above pairs are isolated eigencouples of the linearization. In particular, for each eigencouple we want to build an approximate inverse $\overline{\mathbb{A}}$ of $D\overline{\mathbb{F}}(\nu_{i},u_{i})$ using the construction derived in Section 5.2.

6.3.1 Computation of the needed bounds

As for the proof of the soliton, we need to be able to compute the needed constants introduced in Section 5. Let us fix some $k\in\{1,2,3\}$ and define

[TABLE]

for all $\nu\in\mathbb{R}$ and the norm $H^{l}$ is redefined as explained in Section 5. This definition yields

[TABLE]

In particular, we define $\lambda_{1,k}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\lambda_{1}}{1-\nu_{k}}$, $\lambda_{2,k}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\lambda_{2}}{1-\nu_{k}}$ and $\lambda_{3,k}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\lambda_{3}}{1-\nu_{k}}$, so that

[TABLE]

Recall that $H_{1}=\mathbb{R}\times H^{l}$ and $H_{2}=\mathbb{R}\times L^{2}$ (cf. Section 5). We search for zeros of the following operator $\overline{\mathbb{F}}:H_{1}\to H_{2}$

[TABLE]

To do so, we define $\overline{\mathbb{F}}_{0}:H_{1}\to H_{2}$

[TABLE]

We need to define $\overline{\mathbb{F}}_{0}$ intermediately as explained in Section 5.4. Moreover, as $u_{0}$ and $\tilde{u}$ are close by Theorem 6.6, we use Lemma 5.6 to link $\overline{\mathbb{F}}$ and $\overline{\mathbb{F}}_{0}$. Moreover, define the equivalent periodic operator $F:X_{1}\to X_{2}$

[TABLE]

where $U_{k}$ is the Fourier series representation of $\sqrt{|\Omega_{0}|}u_{k}$ and $X_{1},X_{2}$ are defined in Section 5.

Finally, in order to apply Theorem 4.6,the estimate the passage between $\overline{\mathbb{F}}_{0}$ and $\overline{\mathbb{F}}$ has to be controlled. To do so, Theorem 6.6 provides $\|\mathbb{L}(\tilde{u}-u_{0})\|_{2}\leq r_{0}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}2.27\times 10^{-14}$ and one can use Lemma 5.6. The following Theorem provides the results of the computer-assisted proofs obtained in [47].

Theorem 6.8**.**

Let $(\nu_{1},u_{1}),(\nu_{2},u_{2}),(\nu_{3},u_{3})$ with $\nu_{1}\leq\nu_{2}\leq\nu_{3}$, then for all $1\leq i\leq 3$, there exists $R_{i}$ (given in the table below) such that there exists a unique eigencouple $x_{i}^{*}$ of $\mathbb{L}+D\mathbb{G}(\tilde{u})$ such that $x_{i}^{*}\in\overline{B_{R_{i}}((\nu_{i},u_{i}))}\subset H_{1}=\mathbb{R}\times H^{l}$. Moreover, $\nu_{1},\nu_{2}$ and $\nu_{3}$ are simple eigenvalues.

Remark 6.9**.**

It is easy to check that $(\tilde{u})^{\prime}$ is in the kernel of $\mathbb{L}+D\mathbb{G}(\tilde{u})$. Indeed, one has that

[TABLE]

as $\mathbb{L}\tilde{u}+\lambda_{3}(\tilde{u})^{2}=0$. The previous theorem proves that the eigenvalue [math] is simple, which is not a trivial task in general. Moreover, the fact that $\nu_{1},\nu_{2}$ and $\nu_{3}$ are simple is a direct conclusion of the contraction mapping theorem.

The following result helps in proving that $\nu_{1},\nu_{2},\nu_{3}$ are precisely the first eigenvalues of $\mathbb{L}+D\mathbb{G}(\tilde{u})$.

Lemma 6.10**.**

Let $\nu^{*}\in\mathbb{C}$ and suppose that $\|(\mathbb{L}-\nu^{*}I_{d}+D\mathbb{G}(\tilde{u}))^{-1}\|_{l}\leq\mathcal{C}$. Then $\mathbb{L}-\nu I_{d}+D\mathbb{G}(\tilde{u})$ is invertible for all $\nu\in B_{\frac{1}{\mathcal{C}}}(\nu^{*})\subset\mathbb{C}$ and

[TABLE]

Proof.

The proof is a direct application of Neumann series. ∎

We numerically use Lemma 6.10 to prove Theorem 6.11 below. Indeed, using IntervalArithmetics.jl [55] and the construction of Theorem 3.4, we ensure that $\mathbb{L}-\nu I_{d}+D\mathbb{G}(\tilde{u})$ is non invertible only at isolated values that we found in the previous Table 1. The algorithmic details are exposed in [47].

Theorem 6.11**.**

There is no eigenvalue smaller than $\nu_{1}$ nor in $(\nu_{i},\nu_{i+1})$ for all $i\in\{1,2\}$. In particular, all the other eigenvalues are strictly bigger than $\nu_{3}$.

Proof.

Using the previous Lemma 6.10 we ensure, using interval arithmetics in Julia (cf. [55]) and the construction of Theorem 3.4, that there is no eigenvalue in $(\nu_{i},\nu_{i+1})$ for all $i\in\{1,2\}$. Moreover, as $D\mathbb{G}(\tilde{u})$ is relatively compact with $\mathbb{L}$ and as $\mathbb{L}$ is positive, we know that $\mathbb{L}+D\mathbb{G}(\tilde{u})$ only has finitely many negative eigenvalues. Therefore, if we verify that no eigenvalues are smaller than $\nu_{1}$, then all the other eigenvalues will be strictly larger than $\nu_{3}$. We give a description on the algorithmic proof that provides that $\nu_{1}$ is the smallest eigenvalue.

First, we prove that $\nu_{1}$ is an isolated eigenvalue using a contraction argument. Therefore, we obtain that there exists $r>0$ such that $\nu_{1}$ is the only eigenvalue in $(-r+\nu_{1},\nu_{1}+r)$. Then we consider $\nu^{*}=\nu_{1}-r$, and using the construction of Theorem 3.4 (we build an operator $\mathbb{A}$ approximating the inverse of $\mathbb{L}-\nu^{*}I_{d}+D\mathbb{G}(\tilde{u})$), we numerically prove that $\mathbb{L}-\nu^{*}I_{d}+D\mathbb{G}(\tilde{u})$ is invertible and that the norm of its inverse is bounded by $\mathcal{C}$. Finally Lemma 6.10 gives that $\mathbb{L}-\nu I_{d}+D\mathbb{G}(\tilde{u})$ is invertible for $\nu\in(\nu^{*}-\frac{1}{\mathcal{C}},\nu^{*}+\frac{1}{\mathcal{C}})$.

Now we can repeat the same reasoning by choosing $\tilde{\nu}=\nu^{*}-\frac{1}{\mathcal{C}}$. We iterate the previous steps until we obtain $\|\mathbb{L}-\nu^{*}I_{d}\|_{l,2}>\|D\mathbb{G}(\tilde{u})\|_{l,2}$ for some $\nu^{*}<0$. Then, using that $\|\mathbb{L}-\nu I_{d}\|_{l,2}\geq\|\mathbb{L}-\nu^{*}I_{d}\|_{l,2}$ for all $\nu<\nu^{*}$ (by positivity of $\mathbb{L}$), it yields that $\mathbb{L}-\nu I_{d}+D\mathbb{G}(\tilde{u})$ is invertible for all $\nu\leq\nu^{*}$. In the end it allows to prove that there is no eigenvalue smaller that $\nu_{1}$.

Proving that there is no eigenvalue in $(\nu_{i},\nu_{i+1})$ ($i\in\{1,2\}$) works out identically. ∎

6.4 Proof of stability

Determining rigorously the stability of traveling waves is in general a non-trivial problem. Indeed, in some cases, a spectral theory is not enough to provide a conclusion. The particular case of the Kawahara equation falls into the class of model equations for one-dimensional long nonlinear waves (cf. [71]). Albert in [71] studied the stability of solitary solutions to this specific kind of equations. Other works, such as [64] and [65], also derived some criteria for obtaining the stability of the soliton.

In this section we focus on the results of Albert [71]. The author derived a theorem (see Theorem 3.1 in [71]) that gives necessary conditions to obtain the (orbital) stability of the soliton. Our goal is to prove that these conditions are satisfied in the case of the soliton proved in Theorem 6.6. We first derive a necessary intermediate result.

Proposition 6.12**.**

Let $\mathcal{C}>0$ be such that $\|D\mathbb{F}_{e}(u_{0})^{-1}\|_{l}\leq\mathcal{C}$ and let $r_{0}>0$ and $\tilde{u}\in H^{l}$ defined in Theorem 6.6. If

[TABLE]

where $\kappa$ satisfies (79), then $D\mathbb{F}_{e}(\tilde{u})$ is invertible and

[TABLE]

where $\mathcal{C}_{1}\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\frac{\mathcal{C}}{1-2\kappa\mathcal{C}r_{0}}$.

Proof.

The proof is a simple application of a Neumann series argument combined with the fact that $\|D\mathbb{F}_{e}(\tilde{u})^{-1}\|_{2,l}\leq\mathcal{C}$ and $\|D\mathbb{F}_{e}(\tilde{u})-D\mathbb{F}_{e}(u_{0})\|_{l,2}=\|2\lambda_{3}\mathbb{u}_{0}\|_{l,2}\leq 2\kappa r_{0}$ using (78). ∎

Now that the previous Proposition 6.12 ensures the invertibility of $D\mathbb{F}_{e}(\tilde{u})$, we can state Theorem 3.1 from [71].

Lemma 6.13**.**

(Theorem 3.1 [71]) Let $\tilde{u}\in H^{l}$ be a solution to (77) and assume that

(P1)* $\mathbb{L}+D\mathbb{G}(\tilde{u})$ has a simple negative eigenvalue $\nu_{1}$

(P2) $\mathbb{L}+D\mathbb{G}(\tilde{u})$ has no negative eigenvalue other than $\nu_{1}$

(P3) 0 is a simple eigenvalue of $\mathbb{L}+D\mathbb{G}(\tilde{u})$.*

Now let $\psi\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}D\mathbb{F}_{e}(\tilde{u})^{-1}\tilde{u}$. If

[TABLE]

then $\tilde{u}$ is stable.

Proof.

First, notice that after changing $u\to-u$ and $y\to-y$ in (74), we fall into the required set-up to use the results from [71]. Then, in order to apply Theorem 3.1 in [71], one needs to satisfy the condition (i) stated in the aforementioned article. However, for the Kawahara equation, one can easily prove that this condition is satisfied using Theorem 2.1 (b) in [71]. ∎

We proved conditions $(P1),(P2),(P3)$ in Theorem 6.8. The next proposition gives a rigorous numerical method to prove that $\displaystyle\int_{\mathbb{R}}\tilde{u}\psi<0$ is satisfied.

Proposition 6.14**.**

Let $\tilde{u}$ be the solution obtained in Theorem 6.6. Then, let

[TABLE]

where $A$ is defined in Section 4.3, and let $V=(V_{n})_{n}\in X^{4}_{0}$ be the projection of $V_{0}$ in $X^{4}_{0}$ (using the construction of Theorem 4.2). Finally, we introduce $\epsilon>0$ where

[TABLE]

If there exists $\tau<0$ such that

[TABLE]

then

[TABLE]

In particular, $\tilde{u}$ is stable.

Proof.

We first notice that,

[TABLE]

By construction, we have that $V\in X^{4}_{0}$ and the function representation $\tilde{v}\in H^{l}_{0}(\Omega_{0})$ of $V$ can be extended to a function $v\in H^{l}(\mathbb{R}).$ Now, notice that since $l(\xi)\geq 1$ for all $\xi\in\mathbb{R}$, we have

[TABLE]

Therefore,

[TABLE]

where $\mathcal{C}_{1}$ is defined in Proposition 6.12. Then we obtain that

[TABLE]

where $\|\tilde{u}-u_{0}\|_{2}\leq\|\tilde{u}-u_{0}\|_{l}\leq r_{0}$ (as $l\geq 1$ because $\lambda_{1},\lambda_{2}>0$ in this case) and $r_{0}$ is defined in Theorem 6.6. ∎

Theorem 6.15**.**

The solution $\tilde{u}\in H^{l}_{e}$ obtained in Theorem 6.6 is (orbitally) stable.

Proof.

The proof is accomplished using rigorous numerics in [47] and using Proposition 6.14. In particular, we can choose $\tau\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}-0.048$. ∎

7 Conclusion and Future directions

We presented a new approach to prove strong solutions to semi-linear PDEs in $H^{l}\subset H^{s}(\mathbb{R}^{m})$ via computer-assisted proofs. More specifically, our main result enables a construction of a rigorous inverse of the linearization operator using Fourier series analysis.

Despite Assumptions 1 and 2, the setting is still general enough to be applied to a large class of equations (e.g. KdV equation, Kawahara equation, Swift-Hohenberg equation, etc). However, the relaxation of these assumptions is a future direction of interest and is currently under investigation. We present some ideas to generalize the current method.

First, note that Assumption 2 is a sufficient condition for $\mathbb{G}:H^{l}\to L^{2}$ to be well-defined but it is not necessary. Indeed, in the cases for which $\mathbb{G}:H^{l}\to L^{2}$ is not well-defined, one can apply $(I-\Delta)^{\frac{s}{2}}$ to (5) (if $\psi$ has enough regularity) for some $s\in\mathbb{N}$ big enough and define $\tilde{l}(\xi)\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}l(\xi)(1+|\xi|^{2})^{\frac{s}{2}}$. If $s$ is sufficiently big, then one can use the usual Banach algebra property for Sobolev spaces (see [51]) to obtain the well-definedness of $\mathbb{G}:H^{\tilde{l}}\to L^{2}.$ However, in the current set-up, Assumption 2 is needed for the analysis of Section 3 (cf. Lemma 3.6). Therefore, one could study how the analysis of Section 3 could be generalized under the assumption that $\mathbb{G}:H^{l}\to L^{2}$ is well-defined and smooth, without necessarily imposing Assumption 2.

Another future direction of interest would be the investigation of Assumption 1 to determine if a weaker condition could be used instead. Indeed, as discussed previously in Remark 2.4, the invertibility of $\mathbb{L}$ is needed in the current set-up in order to ensure invertibility of $D\mathbb{F}(u_{0})$. Moreover, we saw in Section 3 that the inverse of $\mathbb{L}$ is used to build a compact operator $D\mathbb{G}(u_{0})\mathbb{L}^{-1}$. Furthermore, it allows to generate an exponential decay that allows us to build an approximate inverse $\mathbb{A}$ (cf. Theorem 3.4). Therefore, Assumption 1 is central in the current analysis to build a computer-assisted approach. It would be interesting to see if similar features could be obtained without the invertibility of $\mathbb{L}$ on the whole $H^{l}$. One way would be to impose restrictions on $H^{l}$ to obtain a subspace on which $\mathbb{L}$ would be invertible.

Moreover, in the case of Kawahara equation, it seems that soliton solutions in $H^{4}(\mathbb{R})$ only exist if $\mathbb{L}$ is invertible (see [68] for instance). This might only be a specific feature of the Kawahara equation but it raises the following question : in the case of an homogeneous PDE, is there any correlation between the existence of solutions in $H^{s}(\mathbb{R}^{m})$ and the invertibility of $\mathbb{L}$ ? In particular, it would be of interest to determine if there is a class of equations for which the invertibility of $\mathbb{L}$ is necessary.

8 Appendix

Lemma 8.1**.**

Let $a,d>0$, then let $n_{1}\in\mathbb{Z}$ and $x_{1},z_{1}\in(-d,d)$. We have

[TABLE]

Similarly,

[TABLE]

Finally, for all $\alpha,\beta\in\mathbb{R}$, we have that

[TABLE]

Proof.

Let us first prove (8.1). Suppose first that $n_{1}>0$, then

[TABLE]

as $2dn_{1}>y_{1}-z_{1}$ for all $y_{1}\in(-d,d)$ as $z_{1}\in(-d,d).$ Therefore, we get

[TABLE]

Now if $n_{1}=0$, we first suppose that $z_{1}\geq x_{1}$, then

[TABLE]

The same reasoning can be applied when $x_{1}\geq z_{1}$ and we obtain (8.1) as a general formula.

In order to prove (8.1), notice that

[TABLE]

where we used the changes of variable $y_{1}\mapsto y_{1}+2dn_{1}$ and $y_{1}\mapsto-y_{1}$. In particular, the previous computations in (8) can be used where $x_{1}$ becomes $-z_{1}$ and $z_{1}$ becomes $-x_{1}$. This proves (8.1).

Finally, to prove (91), we use (8) and take the limit as $d\to\infty$. We treat similarly the case $n_{1}<0$ to conclude the proof. ∎