Rogue periodic waves of the mKdV equation

TL;DR

This paper constructs explicit rogue periodic wave solutions of the focusing mKdV equation using Darboux transformations, analyzes their stability, and compares their magnification factors to those of the NLS equation.

Contribution

It provides explicit formulas for rogue periodic waves of the mKdV equation and analyzes their stability and magnification factors, extending understanding of rogue wave phenomena.

Findings

Rogue dn-periodic wave describes an algebraically decaying soliton over a stable periodic wave.

Rogue cn-periodic wave results from modulation instability and mimics NLS rogue waves.

Magnification factor for the cn-periodic wave remains consistent with the small-amplitude NLS limit.

Abstract

Traveling periodic waves of the modified Korteweg-de Vries (mKdV) equation are considered in the focusing case. By using one-fold and two-fold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation expressed by the Jacobian elliptic functions dn and cn respectively. The rogue dn-periodic wave describes propagation of an algebraically decaying soliton over the dn-periodic wave, the latter wave is modulationally stable with respect to long-wave perturbations. The rogue cn-periodic wave represents the outcome of the modulation instability of the cn-periodic wave with respect to long-wave perturbations and serves for the same purpose as the rogue wave of the nonlinear Schrodinger equation (NLS), where it is expressed by the rational function. We compute the magnification factor for the cn-periodic wave of the mKdV equation and show that it remains the same as in the small-amplitude NLS limit for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the AKNS spectral problem associated with the dn- and cn-periodic waves of the mKdV equation.

Equations

$$i{\psi }_t+{\psi }_{xx}+2(|\psi {|}^2-1)\psi =0,$$

$$\psi (x,t)=1-\frac{4(1+4it)}{1+4x^2+16t^2}.$$

$$u_t+6u^2{u}_x+u_{xxx}=0.$$

$$u_{\mathrm{dn}}(x,t)=\mathrm{dn}(x-ct;k),c=c_{\mathrm{dn}}(k):=2-k^2,$$

$$u_{\mathrm{cn}}(x,t)=k\mathrm{cn}(x-ct;k),c=c_{\mathrm{cn}}(k):=2k^2-1.$$

$$u_{\mathrm{dn}}(x,t)\sim 1-\frac{1}{2}{k}^2{\sin }^2(x-2t)$$

$$u_{\mathrm{cn}}(x,t)\sim k\cos (x+t)$$

$$u_{\mathrm{dn}}(x,t),u_{\mathrm{cn}}(x,t)\to u_{\mathrm{soliton}}(x,t)=\mathrm{sech}(x-t).$$

$$M_{\mathrm{dn}}(k)=2+\sqrt{1-k^2},M_{\mathrm{cn}}(k)=3,k\in [0,1].$$

$$\varphi_{x}=U(\lambda,u)\varphi,\quad U(\lambda,u)=\left(\begin{array}[]{cc}\lambda&u\\ -u&-\lambda\\ \end{array}\right),$$

$$\varphi_{t}=V(\lambda,u)\varphi,\quad V(\lambda,u)=\left(\begin{array}[]{cc}-4\lambda^{3}-2\lambda u^{2}&-4\lambda^{2}u-2\lambda u_{x}-2u^{3}-u_{xx}\\ 4\lambda^{2}u-2\lambda u_{x}+2u^{3}+u_{xx}&4\lambda^{3}+2\lambda u^{2}\\ \end{array}\right).$$

$$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\ i&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),$$

$$U(\lambda ,u)$$

$$V(\lambda ,u)$$

$$\frac{d^{2}u}{d{x}^2}+2u^3=cu,{\left(\frac{du}{dx}\right)}^2+u^4=c{u}^2+d,$$

$$c=4{\lambda }_1^2+2E_0,d=-E_0^2$$

$${\phi }_1^2+{\phi }_2^2=u,{\phi }_1^2-{\phi }_2^2=\frac{1}{2{\lambda }_1}\frac{du}{dx},4{\lambda }_1{\phi }_1{\phi }_2=E_0-u^2.$$

$$\frac{d{\phi }_1}{dx}=\frac{\partial H}{\partial {\phi }_2},\frac{d{\phi }_2}{dx}=-\frac{\partial H}{\partial {\phi }_1},$$

$$H({\phi }_1,{\phi }_2)=\frac{1}{4}({\phi }_1^2+{\phi }_2^2{)}^2+{\lambda }_1{\phi }_1{\phi }_2=\frac{1}{4}{E}_0,$$

$$\frac{du}{dx}=2\left({\phi }_1\frac{d{\phi }_1}{dx}+{\phi }_2\frac{d{\phi }_2}{dx}\right)=2{\lambda }_1({\phi }_1^2-{\phi }_2^2),$$

$$Q(\lambda)=\left(\begin{array}[]{cc}\lambda&\varphi_{1}^{2}+\varphi_{2}^{2}\\ -\varphi_{1}^{2}-\varphi_{2}^{2}&-\lambda\end{array}\right),\quad W(\lambda)=\left(\begin{array}[]{cc}W_{11}(\lambda)&W_{12}(\lambda)\\ W_{12}(-\lambda)&-W_{11}(-\lambda)\end{array}\right),$$

$$W_{11}(\lambda )$$

$$W_{12}(\lambda )$$

$$\frac{d}{dx}W(\lambda )=Q(\lambda )W(\lambda )-W(\lambda )Q(\lambda ),$$

$$\frac{d}{dx}{W}_{12}(\lambda )=2\lambda W_{12}(\lambda )-2({\phi }_1^2+{\phi }_2^2)W_{11}(\lambda )$$

$$W_{12}(\lambda )=\frac{1}{{\lambda }^2-{\lambda }_1^2}\left[\lambda ({\phi }_1^2+{\phi }_2^2)+{\lambda }_1({\phi }_1^2-{\phi }_2^2)\right]=:\frac{(\lambda -\mu )({\phi }_1^2+{\phi }_2^2)}{a(\lambda )},$$

$$a(\lambda ):={\lambda }^2-{\lambda }_1^2\text{mbox}\mathrm{and}\mu :=-{\lambda }_1\frac{{\phi }_1^2-{\phi }_2^2}{{\phi }_1^2+{\phi }_2^2}=-\frac{1}{2u}\frac{du}{dx}.$$

$$\det [W(\lambda )]=-[W_{11}(\lambda ){]}^2-W_{12}(\lambda )W_{21}(\lambda )=-\frac{b(\lambda )}{a(\lambda )}$$

$$b(\lambda ):={\lambda }^2-{\lambda }_1^2-4{\lambda }_1{\phi }_1{\phi }_2-({\phi }_1^2+{\phi }_2^2{)}^2={\lambda }^2-{\lambda }_1^2-E_0.$$

$$[W_{11}(\mu ){]}^2=\frac{b(\mu )}{a(\mu )}.$$

$$\frac{d}{dx}{W}_{12}(\mu )=-\frac{({\phi }_1^2+{\phi }_2^2)}{a(\mu )}\frac{d\mu }{dx}$$

Full text

1 Introduction

Simplest models for nonlinear waves in fluids such as the nonlinear Schrödinger equation (NLS), the Korteweg–de Vries equation (KdV), and the modified Korteweg–de Vries equation (mKdV) have many things in common. First, they appear to be integrable by using the inverse scattering transform method for the same AKNS (Ablowitz–Kaup–Newell–Segur) spectral problem [1]. Second, there exist asymptotic transformations of one nonlinear evolution equation to another nonlinear evolution equation, e.g. from defocusing NLS to KdV and from KdV and focusing mKdV to the defocusing and focusing NLS respectively [36].

Modulation instability of the constant-wave background in the focusing NLS equation has been a paramount concept in the modern nonlinear physics [37]. More recently, spectral instability of the periodic waves expressed by the elliptic functions dn and cn has been investigated in the focusing NLS [18] (see also [23, 25]). Regarding periodic waves in the focusing mKdV equation, it was found that the dn-periodic waves are modulationally stable with respect to the long-wave perturbations, whereas the cn-periodic waves are modulationally unstable [7, 8] (see also [17]).

The outcome of the modulation instability in the focusing NLS equation is the emergence of the localized spatially-temporal patterns on the background of the unstable periodic or quasi-periodic waves (see review in [9]). Such spatially-temporal patterns are known under the generic name of rogue waves [3].

In the simplest setting of the constant wave background, the rogue waves are expressed as rational solutions of the NLS equation. Explicit expressions for such rational solutions have been obtained by using available algebraic constructions such as applications of the multi-fold Darboux transformations [2, 19, 28]. For example, if the focusing NLS equation is set in the form

[TABLE]

then the classical rogue wave up to the translations in $(x,t)$ is given by

[TABLE]

As $|t|+|x|\to\infty$, the rogue wave (1.2) approaches the constant wave background $\psi_{0}(x,t)=1$. On the other hand, at $(x,t)=(0,0)$, the rogue wave reaches the maximum at $|\psi(0,0)|=3$, from which we define the magnification factor of the constant wave background to be $M_{0}=3$. The rogue wave (1.2) was derived by Peregrine [31] as an outcome of the modulational instability of the constant-wave background and is sometimes referred to as Peregrine’s breather.

Rogue waves over nonconstant backgrounds (e.g., the periodic waves or the two-phase solutions) were addressed only recently in the context of the focusing NLS equation (1.1). Computations of such rogue waves rely on the numerical implementation of the Bäcklund transformation to the periodic waves [26] or the two-phase solutions [10] of the NLS. Further analytical work to characterize the general two-phase solutions of the NLS can be found in [35] and in [5, 6].

The purpose of this work is to obtain exact solutions for the rogue waves on the periodic background, which we name here as rogue periodic waves. Computations of such rogue waves are developed by an analytical algorithm with precise characterization of the periodic and non-periodic eigenfunctions of the AKNS spectral problem at the periodic wave. Although our computations are reported in the context of the focusing mKdV equation, the algorithm can be applied to other nonlinear evolution equations associated to the AKNS spectral problem such as the NLS.

Hence we consider the focusing mKdV equation written in the normalized form

[TABLE]

Some particular rational and trigonometric solutions of the mKdV were recently constructed in [15] and discussed in connection to rogue waves of the NLS. In comparison with [15], the novelty of our work is to obtain the rogue periodic waves expressed by the Jacobian elliptic functions and to investigate how these rogue periodic waves generalize in the small-amplitude limit the classical rogue wave (1.2). In particular, we shall compute explicitly the magnification factor for the rogue periodic waves that depends on elliptic modulus of the Jacobian elliptic functions.

There are two particular periodic wave solutions of the mKdV. One solution is strictly positive and is given by the dn elliptic function. The other solution is sign-indefinite and is given by the cn elliptic function. Up to the translations in $(x,t)$ as well as a scaling transformation, the positive solution is given by

[TABLE]

whereas the sign-indefinite solution is given by

[TABLE]

In both cases, $k\in(0,1)$ is elliptic modulus, which defines two different asymptotic limits. In the limit $k\to 0$, we obtain

[TABLE]

and

[TABLE]

which are understood in the sense of the Stokes expansion of the periodic waves. As is well-known [22, 29], the mKdV equation can be reduced asymptotically to the NLS equation in the small-amplitude limit. The cn-periodic wave of the mKdV in the limit $k\to 0$ is reduced to the constant wave background $\psi_{0}$ of the NLS equation (1.1), which is modulationally unstable with respect to the long-wave perturbations. Hence, the cn-periodic wave for the mKdV generalizes the constant wave background of the NLS and inherits modulational instability with respect to long-wave perturbations. The dn-periodic wave has nonzero mean value, which is large enough to make the dn-periodic wave modulationally stable with respect to long-wave perturbations [22].

In the limit $k\to 1$, both Jacobian elliptic functions (1.4) and (1.5) converges to the normalized mKdV soliton

[TABLE]

Very recently, the rogue waves of the mKdV built from a superposition of slowly interacting nearly identical solitons were considered numerically [32] and analytically [33]. It was found in these studies that the magnification factor of the rogue waves built from $N$ nearly identical solitons is exactly $N$.

In our work, we compute the rogue periodic waves for the dn and cn Jacobian elliptic functions with the following algorithm. First, by using the algebraic technique based on the nonlinearization of the Lax pair [11], we obtain the explicit expressions for the eigenvalues $\lambda$ with ${\rm Re}(\lambda)>0$ and the associated periodic eigenfunctions in the AKNS spectral problem associated with the Jacobian elliptic functions. These eigenvalues correspond to the branch points of the continuous bands, when the AKNS spectral problem with the periodic potentials is considered on the real line with the help of the Floquet–Bloch transform [9]. For each periodic eigenfunction, we construct the second, linearly independent solution of the AKNS spectral problem, which is not periodic but linearly growing in $(x,t)$. The latter eigenfunction is expressed in terms of integrals of the Jacobian elliptic functions and hence it is not explicit. Finally, by using the one-fold and two-fold Darboux transformations [24] with the nonperiodic eigenfunctions of the AKNS spectral problem, we obtain the rogue periodic waves. Although the resulting solutions are not explicit, we prove that these solutions approach the dn and cn periodic waves as $|x|+|t|\to\infty$ almost everywhere and that they have maximum at the origin $(x,t)=(0,0)$, where the magnification factors can be computed in the explicit form.

Figure 1 shows the “rogue” dn-periodic wave for $k=0.5$ (left) and $k=0.99$ (right). We write the name of “rogue wave” in commas, because the solution is not a proper rogue wave, the latter one is supposed to appear from nowhere and to disappear without a trace as time evolves [3]. Instead, we obtain a solution that corresponds to a nonlinear superposition of the algebraically decaying soliton of the mKdV [30] and the dn-periodic wave, hence the maximal amplitude is brought by the algebraic soliton from infinity. This outcome of our algorithm is related to the fact that the dn-periodic wave in the mKdV is modulationally stable with respect to the long-wave perturbations [8]. Indeed, it is argued in [4] on several examples involving the constant wave background that the rogue wave solutions exist only in the parameter regions where the constant wave background is modulationally unstable.

Figure 2 shows the rogue cn-periodic wave for $k=0.5$ (left) and $k=0.99$ (right). This solution is a proper rogue wave on the periodic background because it appears from nowhere and disappears without a trace as time evolves. The existence of such rogue periodic wave is related to the fact that the cn-periodic wave in the mKdV is modulationally unstable with respect to the long-wave perturbations [8].

The magnification factors of the rogue periodic waves can be computed in the explicit form:

[TABLE]

It is remarkable that the magnification factor $M_{\rm cn}(k)=3$ is independent of the wave amplitude in agreement with $M_{0}=3$ for the classical rogue wave (1.2) thanks to the small-amplitude asymptotic limit (1.7). At the same time $M_{\rm dn}(k)\in[2,3]$ and $M_{\rm dn}(k)\to 3$ as $k\to 0$ due to the fact that the limit (1.6) gives the same potential to the AKNS spectral problem as the constant wave background $\psi_{0}(x,t)=1$ of the NLS equation (1.1).

In the soliton limit (1.8), $M_{\rm dn}(k)\to 2$ as $k\to 1$ in agreement with the recent results in [32, 33]. Indeed, the “rogue” dn-periodic wave degenerates as $k\to 1$ to the two-soliton solutions constructed of two nearly identical solitons. Such solutions are constructed by the one-fold Darboux transformation from the one-soliton solutions, when the eigenfunction of the AKNS spectral problem is nondecaying (exponentially growing) [27]. Therefore, the magnification factor $M_{\rm dn}(1)=2$ is explained by the weak interaction between two nearly identical solitons. On the other hand, $M_{\rm cn}(1)=3$ is explained by the fact that the rogue cn-periodic wave is built from the two-fold Darboux transformation, hence it degenerates as $k\to 1$ to the three-soliton solutions constructed of three nearly identical solitons [33].

The paper is organized as follows. Section 2 gives details of the periodic eigenfunctions of the AKNS spectral problem associated with the dn and cn Jacobian elliptic functions. The non-periodic functions of the AKNS spectral problem are computed in Section 3. Section 4 presents the general $N$-fold Darboux transformation for the mKdV equation and the explicit formulas for the one-fold and two-fold Darboux transformations. The rogue dn-periodic and cn-periodic waves of the mKdV are constructed in Sections 5 and 6 respectively. Appendix A gives a proof of the $N$-fold Darboux transformations in the explicit form.

Acknowledgement. The authors thank E.N. Pelinovsky for suggesting the problem of computing the rogue periodic waves in the mKdV. J. Chen is grateful to the Department of Mathematics of McMaster University for the generous hospitality during his visit. J. Chen is supported by the National Natural Science Foundation of China (No. 11471072) and the Jiangsu Overseas Research $\&$ Training Programme for University Prominent Young $\&$ Middle-aged Teachers and Presidents (No. 1160690028). D.E. Pelinovsky is supported by the state task of Russian Federation in the sphere of scientific activity (Task No. 5.5176.2017/8.9).

2 Periodic eigenfunctions of the AKNS spectral problem

The mKdV equation (1.3) is obtained as a compatibility condition of the following Lax pair of two linear equations for the vector $\varphi=(\varphi_{1},\varphi_{2})^{t}$:

[TABLE]

and

[TABLE]

The first linear equation (2.1) is referred to as the AKNS spectral problem as it defines the spectral parameter $\lambda$ for a given potential $u(x,t)$ at a frozen time $t$, e.g. at $t=0$. By using the Pauli matrices

[TABLE]

we can rewrite $U(\lambda,u)$ and $V(\lambda,u)$ in (2.1) and (2.2) in the form

[TABLE]

If $u$ is either dn or cn Jacobian elliptic functions (1.4) and (1.5), the potentials are $L$-periodic in $x$ with the period $L=2K(k)$ for dn-functions and $L=4K(k)$ for cn-functions, where $K(k)$ is the complete elliptic integral. If the AKNS spectral problem (2.1) is considered in the space of $L$-periodic functions, then the admissible set for the spectral parameter $\lambda$ is discrete as the AKNS spectral problem has a purely point spectrum.

In the case of periodic or quasi-periodic potentials $u$, the algebraic technique based on the nonlinearization of the Lax pair [11] (see also applications in [12, 13, 14, 20]) can be used to obtain explicit solutions for the eigenfunctions of the AKNS spectral problem related to the particular eigenvalues $\lambda$ with ${\rm Re}(\lambda)>0$. Below we simplify the general method in order to obtain particular solutions of the AKNS spectral problem for the periodic waves in the focusing mKdV equation (1.3). The following two propositions represent the explicit expressions for eigenvalues and periodic eigenfunctions of system (2.1) and (2.2) related to the travelling periodic wave solution of the mKdV.

Proposition 1**.**

Let $u$ be a travelling wave solution of the mKdV equation (1.3) satisfying

[TABLE]

where $c$ and $d$ are real constants parameterized by

[TABLE]

with possibly complex $\lambda_{1}$ and $E_{0}$. Then, there exists a solution $\varphi=(\varphi_{1},\varphi_{2})^{t}$ of the AKNS spectral problem (2.1) with $\lambda=\lambda_{1}$ such that

[TABLE]

In particular, if $u$ is periodic in $x$, then $\varphi$ is periodic in $x$.

Proof.

Following [11], we set $u=\varphi_{1}^{2}+\varphi_{2}^{2}$ and consider a nonlinearization of the AKNS spectral problem (2.1) given by the Hamiltonian system

[TABLE]

which is related to the Hamiltonian function

[TABLE]

where $E_{0}$ is constant in $x$. It follows from (2.9) that $4\lambda_{1}\varphi_{1}\varphi_{2}=E_{0}-u^{2}$. Also note that

[TABLE]

so that all three equations in (2.7) are satisfied by the construction.

Let us introduce

[TABLE]

with

[TABLE]

Then, one can check directly that the Lax equation

[TABLE]

is satisfied for every $\lambda\in\mathbb{C}$ if and only if $(\varphi_{1},\varphi_{2})$ satisfies (2.8). In particular, the $(1,2)$-entry in the above relations yields the equation

[TABLE]

and the representation

[TABLE]

with

[TABLE]

In addition, we note that

[TABLE]

with

[TABLE]

Since $W_{12}(\lambda)$ has a simple zero at $\lambda=\mu$, then

[TABLE]

By substituting (2.11), (2.13), and

[TABLE]

to equation (2.10) and squaring it, we obtain the closed equation on $\mu$:

[TABLE]

Substituting the representation (2.12) yields

[TABLE]

Let $u$ be a solution of the differential equations (2.5) with parameters $c$ and $d$. Substituting (2.5) to (2.15) yields the relations (2.6) between $(c,d)$ and $(\lambda_{1},E_{0})$. Hence, the constraint (2.15) is fulfilled if $u$ satisfies (2.5) with parameters $(c,d)$ satisfying (2.6). ∎

Proposition 2**.**

Let $u$, $\varphi=(\varphi_{1},\varphi_{2})^{t}$, and $\lambda_{1}$ be the same as in Proposition 1. Then $\varphi(x-ct)$ satisfies the linear system (2.2) with $\lambda=\lambda_{1}$ and $u(x-ct)$.

Proof.

By using (2.5), we rewrite the first equation of system (2.2) with $\lambda=\lambda_{1}$ as

[TABLE]

By using (2.7), we note that

[TABLE]

By using (2.6) and the first equation in system (2.1), equation (2.16) becomes

[TABLE]

hence $\varphi_{1}(x-ct)$ is a solution of system (2.1) and (2.2) with $\lambda=\lambda_{1}$ and $u(x-ct)$. Similar computations hold for $\varphi_{2}$ by symmetry from the second equations in systems (2.1) and (2.2). ∎

For the dn Jacobian elliptic functions (1.4), we have $c=2-k^{2}$ and $d=k^{2}-1\leq 0$. Since $u(x)>0$ for every $x\in\mathbb{R}$, the periodic eigenfunction $\varphi=(\varphi_{1},\varphi_{2})^{t}$ in Proposition 1 is real with parameters $E_{0}=\pm\sqrt{1-k^{2}}$ and

[TABLE]

Taking the positive square root of (2.17), we obtain two particular real points

[TABLE]

such that $0<\lambda_{-}(k)<\lambda_{+}(k)<1$ for every $k\in(0,1)$. As $k\to 0$, we have $\lambda_{-}(k)\to 0$ and $\lambda_{+}(k)\to 1$, whereas as $k\to 1$, we have $\lambda_{-}(k),\lambda_{+}(k)\to 1/2$.

For the cn Jacobian elliptic functions (1.5), we have $c=2k^{2}-1$ and $d=k^{2}(1-k^{2})\geq 0$. Since $u(x)$ is sign-indefinite, the periodic eigenfunction $\varphi=(\varphi_{1},\varphi_{2})^{t}$ in Proposition 1 is complex-valued with parameters $E_{0}=\pm ik\sqrt{1-k^{2}}$ and

[TABLE]

Defining the square root of (2.19) in the first quadrant of the complex plane, we obtain

[TABLE]

As $k\to 0$, we have $\lambda_{I}(k)\to i/2$, whereas as $k\to 1$, we have $\lambda_{I}(k)\to 1/2$.

Figure 3 shows the spectral plane of $\lambda$ with the schematic representation of the Floquet–Bloch spectrum for the dn-periodic wave with $k=0.75$ (left) and the cn-periodic wave with $k=0.75$ (right). The branch points $\lambda_{\pm}(k)$ and $\lambda_{I}(k)$ obtained in (2.18) and (2.20) are marked explicitly as the end points of the Floquet–Bloch spectral bands away from the imaginary axis.

3 Non-periodic solutions of the AKNS spectral problem

Here we construct the second linearly independent solution to the AKNS spectral problem (2.1) with $\lambda=\lambda_{1}$ and extend it to satisfy the linear system (2.2). The second solution is no longer periodic in variables $(x,t)$. The following result represents the corresponding solution.

Proposition 3**.**

Let $u$, $\lambda_{1}$, $E_{0}$, and $\varphi=(\varphi_{1},\varphi_{2})^{t}$ be the same as in Proposition 1. Assume that $u(x)^{2}-E_{0}\neq 0$ for every $x$. The second linearly independent solution of the AKNS spectral problem (2.1) with $\lambda=\lambda_{1}$ is given by $\psi=(\psi_{1},\psi_{2})^{t}$, where

[TABLE]

and

[TABLE]

In particular, if $u$ is periodic in $x$, then $\theta$ grows linearly in $x$ as $|x|\to\infty$, so that $\psi_{1}$ and $\psi_{2}$ are not periodic in $x$.

Proof.

Since the AKNS spectral problem (2.1) is related to the traceless matrix, the Wronskian of the two linearly independent solutions $\varphi=(\varphi_{1},\varphi_{2})^{t}$ and $\psi=(\psi_{1},\psi_{2})^{t}$ is independent of $x$. Normalizing it by $2$, we write the relation

[TABLE]

from which the representation (3.1) follows with arbitrary $\theta$. If $u(x)^{2}-E_{0}\neq 0$ for every $x$, then $\varphi_{1}(x)\neq 0$ and $\varphi_{2}(x)\neq 0$ for every $x$. Substituting (3.1) to (2.1), we obtain the following scalar linear differential equation for $\theta$:

[TABLE]

By using relations (2.7), we rewrite it in the equivalent forms:

[TABLE]

Integrating the last equation with the boundary condition $\theta(0)=0$, we obtain (3.2). ∎

Proposition 4**.**

Let $u$, $\lambda_{1}$, $E_{0}$, $\varphi=(\varphi_{1},\varphi_{2})^{t}$, and $\psi=(\psi_{1},\psi_{2})^{t}$ be the same as in Proposition 3. Then, $\psi=(\psi_{1},\psi_{2})^{t}$ expressed by (3.1) satisfies the linear system (2.2) with $\lambda=\lambda_{1}$ and $u(x-ct)$ if $\theta$ is expressed by

[TABLE]

Proof.

By using (2.5), we rewrite the first equation of system (2.2) with $\lambda=\lambda_{1}$ as

[TABLE]

By using (2.7), (3.1), and expressing $\partial_{t}\varphi_{2}$ from the second equation of system (2.2), we obtain from (3.4):

[TABLE]

Let us represent $\theta=-4\lambda_{1}(u^{2}-E_{0})\chi$ so that $\chi$ satisfies

[TABLE]

Hence $\chi(x,t)=-t+f(x-ct)$, where $f$ is obtained from (3.2) in the form

[TABLE]

to yield the representation (3.3). Similar computations hold for $\psi_{2}$ by symmetry from the second equations in systems (2.1) and (2.2). ∎

Note that a more general solution for $\psi=(\psi_{1},\psi_{2})^{t}$ is defined arbitrary up to an addition to the first solution $\varphi=(\varphi_{1},\varphi_{2})^{t}$. However, this addition is equivalent to the arbitrary choice of the lower limit in the integral (3.3), which is then equivalent to the translation in time $t$. Thus, the second linearly independent solution in the form (3.1) and (3.3) is unique up to the translation in $x$ and $t$.

4 One-fold, two-fold, and $N$-fold Darboux transformations

Here we give the explicit formulas for the one-fold and two-fold Darboux transformations for the focusing mKdV equation (1.3), as well as the general formula for the $N$-fold Darboux transformation. Although the formal derivation of the $N$-fold Darboux transformation can be found in several sources, e.g. in book [24] or original papers [21, 34], we find it useful to derive the explicit transformation formulas by using purely algebraic calculations.

By definition, we say that $T(\lambda)$ is a Darboux transformation if

[TABLE]

where $\varphi$ satisfies (2.1)–(2.2) for a particular potential $u$ and $\widetilde{\varphi}$ satisfies (2.1)–(2.2) for a new potential $\widetilde{u}$, which is related to $u$. The transformation formulas between $\varphi$ and $\widetilde{\varphi}$ follow from the Darboux equations

[TABLE]

and

[TABLE]

In many derivations, e.g. in [21, 24, 34], the $N$-fold Darboux transformation is deduced formally from a linear system of equations imposed on the coefficients of the polynomial representation of $T(\lambda)$ without checking all the constraints arising from the Darboux equations (4.2) and (4.3). In order to avoid such formal computations, we give in Appendix A a rigorous derivation of the $N$-fold Darboux transformation in the explicit form and show how the Darboux equations (4.2) and (4.3) are satisfied. Our derivation relies on a particular implementation of the dressing method [38, 39] which was recently reviewed in the context of the cubic NLS equation in [16].

The general $N$-fold Darboux transformation is given by the following theorem.

Theorem 1**.**

Let $u$ be a smooth solution of the mKdV equation (1.3). Let $\varphi^{(k)}=(p_{k},q_{k})^{t}$, $1\leq k\leq N$ be a particular smooth nonzero solution of system (2.1) and (2.2) with fixed $\lambda=\lambda_{k}\in\mathbb{C}\backslash\{0\}$ and potential $u$. Assume that $\lambda_{k}\neq\pm\lambda_{j}$ for every $k\neq j$. Let $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ be a solution of the linear algebraic system

[TABLE]

where $\langle\varphi^{(j)},\varphi^{(k)}\rangle:=p_{j}p_{k}+q_{j}q_{k}$ is the inner vector product. Assume that the linear system (4.4) has a unique solution. Then, $\widetilde{\varphi}^{(k)}=(\widetilde{p}_{k},\widetilde{q}_{k})^{t}$, $1\leq k\leq N$ is a particular solution of system (2.1) and (2.2) with $\lambda=\lambda_{k}$ and the new potential $\widetilde{u}$ given by

[TABLE]

Consequently, $\widetilde{u}$ is a new solution of the mKdV equation (1.3).

The proof of Theorem 1 is given in Appendix A. The following two propositions represent the one-fold and two-fold Darboux transformation formulas deduced from Theorem 1 for $N=1$ and $N=2$ respectively.

Proposition 5**.**

Let $u$ be a smooth solution of the mKdV equation (1.3). Let $\varphi=(p,q)^{t}$ be a particular smooth nonzero solution of system (2.1) and (2.2) with fixed $\lambda=\lambda_{1}\in\mathbb{C}\backslash\{0\}$. Then,

[TABLE]

is a new solution of the mKdV equation (1.3).

Proof.

Solving the linear system (4.4) for $\widetilde{\varphi}=(\widetilde{p},\widetilde{q})^{t}$ yields

[TABLE]

Substituting (4.7) into (4.5) for $N=1$ results in the transformation formula (4.6). ∎

Proposition 6**.**

Let $u$ be a smooth solution of the mKdV equation (1.3). Let $\varphi^{(k)}=(p_{k},q_{k})^{t}$ be a particular smooth nonzero solution of system (2.1) and (2.2) with fixed $\lambda=\lambda_{k}\in\mathbb{C}\backslash\{0\}$ for $k=1,2$ such that $\lambda_{1}\neq\pm\lambda_{2}$. Then,

[TABLE]

is a new solution of the mKdV equation (1.3).

Proof.

The linear system (4.4) is generated by the matrix $A$ with the entries

[TABLE]

For $N=2$, we compute the determinant of this matrix as

[TABLE]

Solving the linear system (4.4) with Cramer’s rule yields the components

[TABLE]

and

[TABLE]

Substituting these formulas to the representation (4.5) with $N=2$ and reordering the similar terms result in the transformation formula (4.8). ∎

5 The “rogue” dn-periodic wave

Here we apply the one-fold Darboux transformation (4.6) to the Jacobian elliptic function dn in (1.4) in order to obtain the “rogue” dn-periodic wave. We write the “rogue” wave in commas, because the corresponding solution is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave, hence the maximal amplitude is brought by the algebraic soliton from infinity. The proper rogue dn-periodic wave does not exist in the mKdV equation (1.3) because the dn-periodic wave is modulationally stable. We note however that very similar solutions to the NLS equation define a proper rogue dn-periodic wave, as is shown numerically in [26].

Let $u$ be the dn periodic wave (1.4), whereas $\varphi=(\varphi_{1},\varphi_{2})^{t}$ be the periodic eigenfunction of the linear system (2.1) and (2.2) with $\lambda=\lambda_{1}$ defined by Propositions 1 and 2. Since the connection formulas (2.7) are satisfied for every $t\in\mathbb{R}$, substituting $p=\varphi_{1}$ and $q=\varphi_{2}$ into the one-fold Darboux transformation (4.6) yields another solution of the mKdV equation in the form

[TABLE]

where $E_{0}=\pm\sqrt{1-k^{2}}$. However, since

[TABLE]

the new solution $\tilde{u}$ to the mKdV equation (1.3) is obtained trivially by the spatial translation of the dn periodic wave on the half-period $\frac{1}{2}L=K(k)$. This computation explains why we need to use the second non-periodic solution $\psi$ instead of the periodic eigenfunction $\varphi$.

Let $u$ be the dn periodic wave (1.4), whereas $\psi=(\psi_{1},\psi_{2})^{t}$ be the non-periodic solution to the linear system (2.1) and (2.2) with $\lambda=\lambda_{1}$ defined by Propositions 3 and 4. Recall that there exist two choices for $\lambda_{1}$ in (2.18). However, for the choice $\lambda_{1}=\lambda_{-}(k)$, we have $E_{0}=\sqrt{1-k^{2}}$ and $u(x)^{2}-E_{0}=0$ for some values of $x$ in $[-K(k),K(k)]$, therefore, the assumption of Proposition 3 is not satisfied. For the choice $\lambda_{1}=\lambda_{+}(k)$, we have $E_{0}=-\sqrt{1-k^{2}}$ and $u(x)^{2}-E_{0}>0$ for every $x$, therefore, the assumption of Proposition 3 is satisfied. Substituting $p=\psi_{1}$ and $q=\psi_{2}$ given by (3.1) into the one-fold Darboux transformation (4.6) with $\lambda_{1}=\lambda_{+}(k)$ and $E_{0}=-\sqrt{1-k^{2}}$ yields another solution of the mKdV equation in the form

[TABLE]

By using relations (2.7) again, we finally write the new solution in the form

[TABLE]

where

[TABLE]

We refer to the exact solution (5.1)–(5.2) as the “rogue” dn periodic wave of the mKdV equation.

If $k=0$, then $u_{\rm dn}(x,t)=1$, $\lambda_{1}=1$, $c=2$, $\theta_{\rm dn}(x,t)=-2(x-6t)$, and

[TABLE]

Although this expression is an analogue of the rogue wave of the NLS on the constant wave background [2, 15], it corresponds to the algebraically decaying soliton of the mKdV [30].

If $k=1$, then $u_{\rm dn}(x,t)={\rm sech}(x-t)$, $\lambda_{1}=\frac{1}{2}$, $c=1$,

[TABLE]

and

[TABLE]

in agreement with the two-soliton solutions of the mKdV for two nearly identical solitons [32, 33].

Next, we show that for every $k\in[0,1)$, there exists a particular line $x=c_{*}t$ with $c_{*}>c$ such that $\theta_{\rm dn}(x,t)$ given by (5.2) remains bounded as $|x|+|t|\to\infty$. This value of $c_{*}$ gives the speed of the algebraically decaying soliton propagating on the dn-periodic wave background. For instance, if $k=0$, then $c_{*}=6>2=c$.

In order to show the claim above, we inspect the expression

[TABLE]

Since the integrand is a positive $L=2K(k)$-periodic function with a positive mean value denoted by $I(k)$, then the expression can be written as

[TABLE]

Therefore, $\theta_{\rm dn}(x,t)$ is bounded at $x=c_{*}t$, where $c_{*}=c+[I(k)]^{-1}>c$.

Except for the line $x=c_{*}t$, the function $\theta_{\rm dn}(x,t)$ given by (5.2) grows linearly in $x$ and $t$ as $|x|+|t|\to\infty$ for every $k\in[0,1)$. Hence the representation (5.1) yields asymptotic behavior

[TABLE]

The maximal value of $u_{\rm dn-rogue}(x,t)$ as $|x|+|t|\to\infty$ except for the line $x=c_{*}t$ coincides with the maximal value of $u_{\rm dn}(x,t)={\rm dn}(x-ct;k)$.

For $t=0$, $u_{\rm dn}(x,0)$ is even in $x$, $\theta_{\rm dn}(x,0)$ is odd in $x$, hence $u_{\rm dn-rogue}(x,0)$ is even in $x$. The maximal value of $u_{\rm dn}(x,0)$ occurs at $u_{\rm dn}(0,0)=1$. Since $u_{\rm dn-rogue}(x,0)$ is even in $x$, then $x=0$ is an extremal point of $u_{\rm dn-rogue}(x,0)$. Moreover, $\partial_{x}^{2}u_{\rm dn-rogue}(0,0)<0$, which follows from the expansions $u_{\rm dn}(x,0)=1-\frac{1}{2}k^{2}x^{2}+\mathcal{O}(x^{4})$, $\theta_{\rm dn}(x,0)=-4\lambda_{1}(1+\sqrt{1-k^{2}})^{-1}x+\mathcal{O}(x^{3})$, and

[TABLE]

Hence $x=0$ is the point of maximum of $u_{\rm dn-rogue}(x,0)$. Defining the magnification number as

[TABLE]

we obtain the expression in (1.9). The value $M_{\rm dn}(k)$ corresponds to the amplitude of the algebraically decaying soliton propagating on the background of the dn-periodic wave.

6 The rogue cn-periodic wave

Here we apply the one-fold and two-fold Darboux transformations (4.6) and (4.8) to the Jacobian elliptic function cn in (1.5) in order to obtain the rogue cn-periodic wave. This is a proper rogue cn-periodic wave because the cn-periodic wave is modulationally unstable.

Let $u$ be the cn periodic wave (1.5), whereas $\varphi=(\varphi_{1},\varphi_{2})^{t}$ be the periodic solution to the linear system (2.1) and (2.2) with $\lambda=\lambda_{1}$ defined by Propositions 1 and 2. Without loss of generality, we choose $\lambda_{1}=\lambda_{I}(k)$, where $\lambda_{I}(k)$ is given by (2.20), so that $E_{0}=-ik\sqrt{1-k^{2}}$. Since the periodic solution $\varphi$ is complex, the one-fold Darboux transformation (4.6) produces a complex-valued solution to the mKdV, hence we should use the two-fold Darboux transformation (4.8).

By virtue of relations (2.7), substituting $(p_{1},q_{1})=(\varphi_{1},\varphi_{2})$ with $\lambda_{1}=\lambda_{I}$ and $(p_{2},q_{2})=(\overline{\varphi}_{1},\overline{\varphi}_{2})$ with $\lambda_{2}=\overline{\lambda}_{I}$ to the two-fold Darboux transformation (4.8) yields another solution of the mKdV equation in the form

[TABLE]

where the first-order invariant in (2.5) is used in the second identity with $c=2k^{2}-1$ and $d=k^{2}(1-k^{2})$. Thus, the new solution $\tilde{u}$ in the two-fold transformation (4.8) is trivially related to the previous solution $u$ if the functions $(p_{1},q_{1})$ and $(p_{2},q_{2})$ are periodic.

Let us now consider the non-periodic solution $\psi=(\psi_{1},\psi_{2})^{t}$ to the linear system (2.1) and (2.2) with $\lambda=\lambda_{I}$. The assumption of Proposition 3 is satisfied because $E_{0}=-ik\sqrt{1-k^{2}}\neq 0$ for $k\in(0,1)$ and $u(x)^{2}-E_{0}\neq 0$ for every $x$. Therefore, the non-periodic solution $\psi$ in Propositions 3 and 4 is well-defined. Substituting $(p_{1},q_{1})=(\psi_{1},\psi_{2})$ with $\lambda_{1}=\lambda_{I}$ and $(p_{2},q_{2})=(\overline{\psi}_{1},\overline{\psi}_{2})$ with $\lambda_{2}=\overline{\lambda}_{I}$ into the two-fold Darboux transformation (4.8) yields another solution of the mKdV in the form

[TABLE]

where

[TABLE]

By using relations (2.7) and (2.20), we finally write the new solution in the form

[TABLE]

where

[TABLE]

and

[TABLE]

We refer to the exact solution (6.1)–(6.2) as the rogue cn periodic wave of the mKdV equation.

As $k\to 0$, then $u_{\rm cn}(x,t)\to 0$, $\lambda_{I}\to i/2$, $\theta_{\rm cn}(x,t)\to 0$, and $u_{\rm cn-rogue}(x,t)\to 0$. Although the limit is zero, one can derive asymptotic expansions at the order of $\mathcal{O}(k)$ which recovers the rogue wave of the NLS equation (1.1), according to the asymptotic transformation of the focusing mKdV to the focusing NLS in the small-amplitude limit [22]. The rouge cn-periodic wave generalizes the rogue wave (1.2) on the constant wave background.

As $k\to 1$, then $u_{\rm cn}(x,t)\to{\rm sech}(x-t)$, $\lambda_{I}\to 1/2$, and it may first seem that the second term in (6.1) vanishes. However, $G_{1}=\mathcal{O}(1-k^{2})$ and $G_{2}=\mathcal{O}(1-k^{2})$, hence a non-trivial limit exists to yield a three-soliton solution to the mKdV with three nearly identical solitons [33].

Let us inspect the expression

[TABLE]

For every $k\in(0,1)$, the imaginary part in the integrand is a negative $L=4K(k)$-periodic function with a negative mean value. It is only bounded on the line $x=ct$, however, the real part of the last term in the expression grows linearly in $t$. Therefore, for every $k\in(0,1)$, $|\theta_{\rm cn}(x,t)|$ grows linearly in $x$ and $t$ as $|x|+|t|\to\infty$ everywhere on the $(x,t)$ plane. Hence the representation (6.1) yields the asymptotic behavior

[TABLE]

where the first-order invariant in (2.5) is used for the last identity with $c=2k^{2}-1$ and $d=k^{2}(1-k^{2})$. The maximal value of $u_{\rm cn-rogue}(x,t)$ as $|x|+|t|\to\infty$ coincides with the maximal value of $u_{\rm cn}(x,t)=k{\rm cn}(x-ct;k)$.

For $t=0$, $u_{\rm cn}(x,0)$ is even in $x$, $\theta_{\rm cn}(x,0)$ is odd in $x$, hence $u_{\rm cn-rogue}(x,0)$ is even in $x$. The maximal value of $u_{\rm cn}(x,0)$ occurs at $u_{\rm cn}(0,0)=k$. Since $u_{\rm cn-rogue}(x,0)$ is even in $x$, then $x=0$ is an extremal point of $u_{\rm cn-rogue}(x,0)$. Moreover, $\partial_{x}^{2}u_{\rm cn-rogue}(0,0)<0$, which follows from the expansions $u_{\rm cn}(x,0)=k-\frac{1}{2}kx^{2}+\mathcal{O}(x^{4})$, $\theta_{\rm cn}(x,0)=-4\lambda_{I}(k^{2}-ik\sqrt{1-k^{2}})x+\mathcal{O}(x^{3})$, and

[TABLE]

Hence $x=0$ is the point of maximum of $u_{\rm cn-rogue}(x,0)$. Defining the magnification number as

[TABLE]

we obtain the expression in (1.9). The magnification factor is independent of the amplitude of the cn-periodic wave.

Appendix A Proof of $N$-fold Darboux transformation

Here we prove Theorem 1 with explicit algebraic computations. The Darboux transformation matrix $T(\lambda)$ in (4.1) is sought in the following explicit form:

[TABLE]

where the sign $\otimes$ denotes the outer vector product and $I$ denotes an identity $2\times 2$ matrix.

We note that $\varphi^{(k)}\in{\rm ker}(T_{k})$ and $\widetilde{\varphi}^{(k)}\in{\rm ran}(T_{k})$. It is assumed in Theorem 1 that $\varphi^{(k)}=(p_{k},q_{k})^{t}$, $1\leq k\leq N$ is a particular smooth nonzero solution to system (2.1) and (2.2) with fixed $\lambda=\lambda_{k}\in\mathbb{C}\backslash\{0\}$ satisfying $\lambda_{k}\neq\pm\lambda_{j}$ for every $k\neq j$, whereas $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ is a unique solution of the linear algebraic system (4.4). Deeper in the proof, we will be able to show that $\widetilde{\varphi}^{(k)}=(\widetilde{p}_{k},\widetilde{q}_{k})^{t}$, $1\leq k\leq N$ is a particular solution to system (2.1) and (2.2) with $\lambda=\lambda_{k}$ and new potential $\widetilde{u}$ given by the transformation formula (4.5).

First, let us show that the two lines in the definition (4.5) are identical. Let us define entries of the matrix $A$ by (4.9). Each entry is finite, moreover, $A_{jk}=A_{kj}$. The linear system (4.4) can be split into two parts as follows

[TABLE]

Thanks to the symmetry of $A$, we obtain from (A.2):

[TABLE]

This proves that the two lines in the definition (4.5) are identical. For further use, let us also derive another relation from the system (A.2):

[TABLE]

Next, we show validity of the Darboux equation (4.2) under the transformation formula (A.1). Substituting (A.1) to (4.2) yields the following equations at the simple poles

[TABLE]

and the following equation at the constant term

[TABLE]

Equation (A.6) yields (4.5) due to representation (A.1).

Let us show that equations (A.5) are satisfied if $\varphi^{(k)}$ solves (2.1) with $\lambda=\lambda_{k}$ and $u$, whereas $\widetilde{\varphi}^{(k)}$ solves (2.1) with $\lambda=\lambda_{k}$ and $\widetilde{u}$. Recall that $\sigma_{1}\sigma_{3}=-\sigma_{3}\sigma_{1}$ and $\sigma_{1}\sigma_{1}=\sigma_{3}\sigma_{3}=I$. Substituting (A.1) to both sides of (A.5) yields

[TABLE]

and

[TABLE]

hence equation (A.5) is satisfied.

We show now that if $\{\varphi^{(k)}\}_{1\leq k\leq N}$ solve (2.1) with $\{\lambda_{k}\}_{1\leq k\leq N}$ and $u$ and $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ are obtained from the linear algebraic system (4.4), then $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ solve (2.1) with $\{\lambda_{k}\}_{1\leq k\leq N}$ and $\widetilde{u}$. We note from the linear system (2.1) that

[TABLE]

Differentiating (4.4) in $x$ and substituting (2.3) and (A.7) yield

[TABLE]

where the last equality is due to the transformation formula (4.5). Thus, if the linear system (4.4) is assumed to admit a unique solution, then $\widetilde{\varphi}^{(k)}$ solves (2.1) with $\lambda=\lambda_{k}$ and $\widetilde{u}$.

It remains to show validity of the Darboux equation (4.3) under the transformation formula (A.1). Substituting (A.1) to (4.3) yields the following equations at the simple poles

[TABLE]

the same equation (A.6) at $\lambda^{2}$ and the following two equations at $\lambda^{1}$ and $\lambda^{0}$ respectively:

[TABLE]

and

[TABLE]

Let us show that equations (A.9) are satisfied if $\varphi^{(k)}$ solves (2.2) with $\lambda=\lambda_{k}$ and $u$, whereas $\widetilde{\varphi}^{(k)}$ solves (2.2) with $\lambda=\lambda_{k}$ and $\widetilde{u}$. Substituting (A.1) to both sides of (A.9) yields

[TABLE]

and

[TABLE]

hence equation (A.9) is satisfied.

In order to show the validity of equation (A.10), we differentiate (A.6) in $x$ and substitute (A.5) to obtain

[TABLE]

Substituting (A.12) into (A.10) yields a simplified form of the equation:

[TABLE]

Further substituting (A.6) into (A.13) yields

[TABLE]

The validity of equation (A.14) is satisfied thanks again to equation (A.6):

[TABLE]

Hence, equation (A.10) is satisfied.

In order to show the validity of equation (A.11), we differentiate (A.12) in $x$ and substitute (A.5) to obtain

[TABLE]

Substituting (A.6) and (A.16) into (A.11) yields a simplified form of the equation:

[TABLE]

The last term in the left-hand side of (A.17) is identically zero thanks to equation (A.14) after multiplication by $\sigma_{1}$ on the right. Multiplication of equation (A.14) by $\sigma_{3}$ on the right allows us to group the terms containing $u_{x}$ and $\widetilde{u}_{x}$. As a result, we rewrite (A.17) in the equivalent form

[TABLE]

Multiplying (A.12) by $\sigma_{1}$ from the left and from the right, we obtain

[TABLE]

and

[TABLE]

from which one can rewrite (A.18) in the equivalent form

[TABLE]

which is satisfied thanks to equation (A.15). Hence, equation (A.11) is satisfied.

Finally, we show that if $\{\varphi^{(k)}\}_{1\leq k\leq N}$ solve (2.2) with $\{\lambda_{k}\}_{1\leq k\leq N}$ and $u$ and $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ are obtained from the linear algebraic system (4.4), then $\{\widetilde{\varphi}^{(k)}\}_{1\leq k\leq N}$ solve (2.2) with $\{\lambda_{k}\}_{1\leq k\leq N}$ and $\widetilde{u}$. We note from the linear system (2.2) that

[TABLE]

Differentiating (4.4) in $t$ and substituting (2.4) and (A.19) yield

[TABLE]

The terms proportional to $4\lambda_{j}^{2}$ cancel out due to the same relation (A.8). The terms proportional to $2\lambda_{j}$ cancel out if the following relation is true:

[TABLE]

The other $\lambda_{j}$-independent terms cancel out if the following relation is true:

[TABLE]

Provided equations (A.21) and (A.22) are satisfied, the right-hand side of equation (A.20) is zero. If the linear system (4.4) is assumed to admit a unique solution, then $\widetilde{\varphi}^{(k)}$ solves (2.2) with $\{\lambda_{k}\}_{1\leq k\leq N}$ and $\widetilde{u}$.

Finally, we show validity of equations (A.21) and (A.22). In order to show (A.21), we first obtain the relation

[TABLE]

in addition to the relation (A.7). Then, we differentiate (A.8) in $x$, substitute (2.3), (A.7), and (A.23), and obtain

[TABLE]

where the relation (A.8) was used to cancel the $\lambda_{j}$ term. By using the transformation formulas (4.5), we verify that

[TABLE]

This allows us to simplify (A.24) to the form

[TABLE]

Substituting (A.26) to (A.21) yields the following equation

[TABLE]

Thanks to the relations (A.8) and (A.25), equation (A.27) is satisfied, and so is equation (A.21).

In order to show (A.22), we first obtain the relations

[TABLE]

and

[TABLE]

Then, we differentiate (A.26) in $x$, substitute (2.3), (A.7), (A.23), (A.28), and (A.29), and obtain

[TABLE]

where the relation (A.26) was used to cancel the $\lambda_{j}$ term. Substituting (A.30) into (A.22) and using (A.8) and (A.25) yield

[TABLE]

Substituting (A.26) to (A.31) yields

[TABLE]

By using the relations (A.3) and explicit computations, we obtain

[TABLE]

[TABLE]

and

[TABLE]

Substituting (A.33), (A.34), and (A.35) to (A.32) cancel all terms thanks to the relations (A.4) and (A.8). Therefore, equation (A.32) is satisfied, and so is equation (A.22).