Traveling kinks in an infinite array of weakly coupled pendula

TL;DR

This paper proves the existence of traveling kink solutions in an infinite array of weakly coupled pendula using a perturbation approach from the anti-continuum limit.

Contribution

It introduces a novel application of perturbation methods to establish heteroclinic traveling waves in coupled pendula systems.

Findings

Existence of heteroclinic traveling waves proven

Perturbation method from anti-continuum limit applied

Framework applicable to similar coupled oscillator systems

Abstract

We prove the existence of heteroclinic traveling waves (kinks) in an infinite array of weakly coupled pendula. Our approach is to apply a perturbation argument from the anti-continuum limit.

Equations

$${\ddot{v}}_n-k(v_{n+1}-2v_n+v_{n-1})+\sin (v_n)=0,n\in \mathbb{Z}.$$

$$c^2\frac{d^{2}v}{d{z}^2}-k[v(z+1)-2v(z)+v(z-1)]+\sin (v)=0.$$

$${\ddot{v}}_n-k(v_{n+1}-2v_n+v_{n-1})+2(v_n-v_n^3)=0,n\in \mathbb{Z},$$

$$u^{\prime \prime }-ϵ\sum _{k=-\infty }^{\infty }{a}_{k}u(z-k)+f(u)=0,z\in \mathbb{R},$$

$$\underset{z\to -\infty }{\lim }u(z)=0,\underset{z\to \infty }{\lim }u(z)=1.$$

$$F(1)=0,F(u)\ne 0\forall u\in (0,1)$$

$$u^{\prime \prime }+f(u)=0.$$

$$||u_{ϵ}-u_0|{|}_{H^2(\mathbb{R})}\le Cϵ$$

$$u=\frac{v+\pi }{2\pi }\text{and}f(u)=\frac{1}{2\pi }\sin (2\pi u-\pi ).$$

$$u=\frac{v+1}{2}\text{and}f(u)=(2u-1)-(2u-1{)}^3,$$

$${\ddot{u}}_n-ϵ\sum _{k=-\infty }^{\infty }{a}_k{u}_{n-k}+f(u_n)=0,n\in \mathbb{Z},$$

$$H(\text{p},\text{u})=\sum _n\left(\frac{1}{2}{p}_n^2+ϵ\sum _m{a}_{n-m}(u_n-u_m{)}^2+F(u_n)\right),$$

$$(\varphi ,\psi )\equiv {\int }_{\mathbb{R}}\varphi \psi dz;\varphi \perp \psi \iff (\varphi ,\psi )=0.$$

$$\Delta u:=\sum _{k=-\infty }^{\infty }{a}_{k}u(z-k).$$

$$||\Delta u|{|}_{L^2}\le C||u|{|}_{L^2}(C\text{independent of }u),$$

$$(\Delta u,v)=(u,\Delta v).$$

$$\underset{z\to -\infty }{\lim }u(z)=u(-\infty )\in \mathbb{R}\text{and}\underset{z\to \infty }{\lim }u(z)=u(\infty )\in \mathbb{R},$$

$$\Delta u\in L^2(\mathbb{R})and(\Delta u,u^{\prime })=0.$$

$$u_0^{\prime }(z),|u_0^{\prime \prime }(z)|,|u_0^{\prime \prime \prime }(z)|\le C{e}^{-c|z|},z\in \mathbb{R}.$$

$$L^0\varphi ={\varphi }^{\prime \prime }+f^{\prime }(u_0(z))\varphi$$

$$L^0\varphi =g.$$

$$||\varphi |{|}_{H^2}\le C||g|{|}_{L^2}$$

$$L^0\varphi =ϵ\Delta \varphi +N(\varphi )+ϵ\Delta u_0=E(\varphi )$$

$$N(\varphi )=-f(u_0+\varphi )+f(u_0)+f_u(u_0)\varphi .$$

$$L^0\psi =-b(\varphi )u_0^{\prime }+ϵ\Delta \varphi +N(\varphi )+ϵ\Delta u_0$$

$$b(\varphi )=\frac{1}{||u_0^{\prime }|{|}_{L^2}^2}\left(ϵ\Delta \varphi +N(\varphi )+ϵ\Delta u_0,u_0^{\prime }\right).$$

$$B_{ϵ}=\{\varphi \in H^2(\mathbb{R})\cap \{u_0^{\prime }{\}}^{\perp }:||\varphi |{|}_{H^2}\le Mϵ\}$$

$$||\psi |{|}_{H^2}$$

$$|N(\varphi )|\le CMϵ|\varphi |\text{and}|N({\varphi }_1)-N({\varphi }_2)|\le CMϵ|{\varphi }_1-{\varphi }_2|$$

$$||\psi |{|}_{H^2}$$

Full text

Introduction and main result

The model

The following infinite system of linearly coupled ODEs describes the motion of an array of pendula each of which is coupled to its nearest neighbors by a torsional spring with a coupling coefficient $k$:

[TABLE]

More precisely, $v_{i}$ represents the angle formed by the $i$th pendulum with the vertical axis (assuming physical units have been scaled appropriately). We refer the reader to [KT, Le, Sc] for more details on the physical background of the problem.

The system (1) is known as the discrete sine-Gordon equation and also serves as a model of arrays of Josephson junctions [IS], or as a dynamical Frenkel-Kontorova model of electrons in a crystal lattice [BK].

Traveling wave solutions

We shall construct solutions of the above equation in the form of traveling waves (cf. [CMS]). To this end, we let $z=n$, write $v_{n}(t)=v(z,t)$, and seek a solution of the form $v(z,t)=v(\xi)=v(z-ct)$ satisfying the equation

[TABLE]

Known results

In [S] explicit kink solutions to the above equation were given for a certain nonlinearity $f$ (in place of the sine) which satisfies (H1) below. Periodic traveling wave solutions to (1) have been shown to exist recently in [FR1, FR2, SL] in the strong coupling regime (i.e. when $k\gg 1$). This was achieved in [FR1] using techniques from dynamical systems; in [FR2] using variational and topological techniques; in [SL] using a fixed point argument. However, as is pointed out in [FR1], kink solutions to (2) connecting $-\pi$ to $\pi$ should not exist in this regime. Nevertheless, such solutions were constructed variationally in [KZ] provided that $c$ is sufficiently large. The question of persistence of kink solutions in the continuum limit of (1) and (3) was discussed in [ACR, DKY, IP, OPB, SZE]. We also refer to [FR3, KKCR] for further results on the existence of localized structures in long range interaction lattices (stationary or traveling).

The main result

In the current paper, for given $c\neq 0$, we study heteroclinic waves for sufficiently small $k$, which from now on we will call $\varepsilon$. We will also consider a more general class of nonlinearities $f$ satisfying (H1) below, covering both the discrete sine-Gordon equation as well as the important $\phi^{4}$ model [BeK]:

[TABLE]

see Remark 1 below. Moreover, motivated from [BCC], we will allow infinite range and not just nearest neighbor or finite length interaction, although those are included as special cases.

The equation we will be dealing with is

[TABLE]

together with the conditions

[TABLE]

Here $\epsilon\geq 0$ is small, and we assume that

(H1) $f\in C^{2}(\mathbb{R}),\ f(0)=f(1)=0,\ \ f^{\prime}(0),\ f^{\prime}(1)<0;$

[TABLE]

where $F(u)=\int_{0}^{u}f(s)ds.$

(H2) $\sum_{k=-\infty}^{\infty}a_{k}=0,\ a_{0}<0,\ a_{k}=a_{-k},\ \textrm{and}\ \sum_{k\geq 1}|a_{k}|k^{2}<\infty.$

When $\epsilon=0$, that is in the so called anti-continuum limit [MA], equation (4) becomes

[TABLE]

Under the hypotheses (H1), the above equation has a heteroclinic solution $u_{0}$ satisfying (5) (see [Ar]).

Our result is

Theorem 1**.**

If $\epsilon>0$ is sufficiently small, then there exists a solution $u_{\epsilon}$ of (4) such that

[TABLE]

($C>0$ is a constant independent of $\epsilon$).

Remark 1**.**

The choice of the roots of $f$ to be [math] and $1$ is made for convenience purposes only and causes no loss of generality. For instance, the traveling kink problem (2) can be embedded in our framework by plainly letting

[TABLE]

Similarly, the corresponding change of variables for (3) is

[TABLE]

(keep in mind the first assumption in (H2)).

Method of proof

To prove this we adapt a technique from an earlier paper of ours (cf. [AFFS], but see also [dPK] for a related idea). We use two important properties:

(i) Nondegeneracy of $u_{0}$ The operator obtained by linearizing the left-hand side of (6) at $u_{0}$ has [math] as a simple isolated eigenvalue, the remaining spectrum being in the open left half-plane ([He], Section 5.4).

(ii) Hamiltonian form of the problem The equation (4) arises when seeking traveling waves to the lattice equation

[TABLE]

which comes from the Hamiltonian on $\ell^{2}\times\ell^{2}$ defined by

[TABLE]

where $\textbf{p}=(\dot{u}_{n})\in\ell^{2}$ and $\textbf{u}=({u}_{n})\in\ell^{2}$.

As is explained in [SZ], the persistence of heteroclinic or homoclinic orbits of Hamiltonian systems under Hamiltonian perturbations is a delicate issue, as the Melnikov integral vanishes. An analogous difficulty also arises in the lattice setting at hand, as we will discuss in Section 2.

Outline of the paper

In Section 1 we present the proof of Theorem 1, in Section 2 we make some comments on the application of the standard Lyapunov-Schmidt reduction, and in the Appendix’s we prove some technical lemmas.

Notation

In what follows, $||\cdot||_{L^{2}}$, $||\cdot||_{L^{\infty}}$, and $||\cdot||_{H^{i}},\ (i=1,2)$ denote the norms of the spaces $L^{2}(\mathbb{R})$, $L^{\infty}(\mathbb{R})$, and $H^{i}(\mathbb{R})$, respectively. Also,

[TABLE]

Unless specified otherwise $C/c$ denotes a large/small positive constant independent of $\epsilon>0$ whose value will change from line to line. In many cases we will not explicitly write the obvious dependence of functions on $\epsilon$.

1. Proof of Theorem 1

1.1. Properties of the linear operator $\Delta$

Consider the linear operator $\Delta$ defined via

[TABLE]

Then one can verify that if $u,v\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\cap C(\mathbb{R})$, then

[TABLE]

and

[TABLE]

We also have the following lemma whose proof will be given in Appendix A.

Lemma 1**.**

If $u\in C^{1}(\mathbb{R})$, $u^{\prime}\in L^{2}(\mathbb{R})$;

[TABLE]

then

[TABLE]

1.2. Properties of the heteroclinic $u_{0}$

It is well known (see [Ar], [HK]) that $u_{0}$ is the unique (up to translation) solution of (6), (5). Furthermore $u_{0}^{\prime}>0$, $u_{0}$ approaches its limits exponentially and

[TABLE]

The linear operator $L^{0}$ with $D(L^{0})=H^{2}(\mathbb{R})$ and

[TABLE]

is self-adjoint in $L^{2}(\mathbb{R})$ and $\sigma(-L^{0})\subseteq\{0\}\cup[c,\infty)$ with [math] a simple eigenvalue corresponding to $u_{0}^{\prime}$. (Note that since $f^{\prime}(0),\ f^{\prime}(1)<0$ then $\sigma_{ess}(-L^{0})\subseteq[c,\infty)$; thus the only thing left to prove is that [math] is the principal eigenvalue which follows from $u_{0}^{\prime}>0$.)

These properties imply the following important proposition.

Proposition 1**.**

Let $g\in L^{2}(\mathbb{R})$ with $g\bot u_{0}^{\prime}$, then there exists a unique $\phi\in H^{2}(\mathbb{R})$ with $\phi\bot u_{0}^{\prime}$ such that

[TABLE]

Furthermore, we have

[TABLE]

with $C$ independent of g.

1.3. The perturbation argument

We search for a solution of (4) in the form $u_{\epsilon}=u_{0}+\phi_{\epsilon}$ with $\phi_{\epsilon}\in H^{2}(\mathbb{R})$ and $\phi_{\epsilon}\bot u_{0}^{\prime}.$ Then, the fluctuation $\phi_{\epsilon}$ must satisfy

[TABLE]

with

[TABLE]

We note that $\left(E(\phi_{\epsilon}),u_{0}^{\prime}\right)=0$ holds. However, if iterations of the form $L^{0}\phi_{n+1}=E(\phi_{n})$ were to be performed, for capturing the desired $\phi_{\epsilon}$ in the limit $n\to\infty$, the iteration $\phi_{n}$ may not satisfy this orthogonality condition which is necessary for solving for $\phi_{n+1}\in H^{2}(\mathbb{R})$. To deal with this issue, at each step of the iteration we will project $E(\phi_{n})$ to $\{u_{0}^{\prime}\}^{\bot}$ and then solve for the corresponding $\phi_{n+1}$.

We thus define a mapping ${T}:H^{2}(\mathbb{R})\cap\{u_{0}^{\prime}\}^{\bot}\to H^{2}(\mathbb{R})\cap\{u_{0}^{\prime}\}^{\bot}$ via ${T}(\phi)={\psi}$ where

[TABLE]

and

[TABLE]

Note that ${T}$ is well defined via Proposition 1 since the right hand side of (9) is orthogonal to $u_{0}^{\prime}$. (Note also that by Lemma 1 we have $\Delta u_{0}\in L^{2}(\mathbb{R}).$)

Let

[TABLE]

with $M$ a positive constant independent of $\epsilon>0$ to be determined later. We will show that there exists a large $M>0$ such that, provided $\epsilon>0$ is sufficiently small, ${T}$ maps $B_{\epsilon}$ into itself and is a contraction. Let $\phi\in B_{\epsilon}$, then via (9), (10) and Proposition 1,

[TABLE]

The first and third term will be estimated from (7) and Lemma 1 respectively. To estimate the nonlinear term $N(\phi)$, we first recall the embedding $||\phi||_{L^{\infty}}\leq C||\phi||_{H^{1}}$ for every $\phi\in H^{1}(\mathbb{R})$. Hence, setting $C=\sup_{|s|\leq 2}|f^{\prime\prime}(s)|$, we have

[TABLE]

pointwise for all $\phi,\phi_{1},\phi_{2}\in B_{\epsilon}$. Thus, (LABEL:eq8) yields

[TABLE]

Choosing a large $M$ (say $M=2C$), then $||{\psi}||_{H^{2}}\leq M\epsilon$ provided $\epsilon>0$ is sufficiently small, i.e. ${T}:B_{\epsilon}\to B_{\epsilon}$. Similarly we can show that ${T}$ is a contraction in the $H^{2}$ norm.

Since $B_{\epsilon}$ is closed with respect to this norm, the Banach fixed point theorem gives us a fixed point $\phi_{*}\in B_{\epsilon}$ of ${T}$. Then

[TABLE]

satisfies

[TABLE]

for some $b_{\epsilon}\in\mathbb{R}$ ($b_{\epsilon}=b(\phi_{*}^{\epsilon})$). Multiplying (14) by $u^{\prime}=u_{\epsilon}^{\prime}$ and integrating over $\mathbb{R}$ yields

[TABLE]

Since $\phi_{*}\in H^{2}(\mathbb{R})$, we have that $u(-\infty)=0$, $u(\infty)=1$ and $u^{\prime}(\pm\infty)=0$. The left hand side of the above equation is 0 (see Lemma 1 and recall that $F(1)=0$) and we get that

[TABLE]

This implies that $b_{\epsilon}=0$ since

[TABLE]

provided $\epsilon>0$ is sufficiently small.

Therefore $u$ given by (13) is a solution of (4) satisfying the estimate of Theorem 1, thereby completing the proof.

Remark 2**.**

If $f(u+1/2)$ is odd, as it would be the case for applications to (2) and (3), then the proof of Theorem 1 becomes much simpler. Indeed, we can apply our fixed point argument restricted to the space of functions such that $\phi-1/2$ is odd, without the need to introduce the projection operator. Keep in mind that $u_{0}-\frac{1}{2}$ would be odd and $u_{0}^{\prime}$ would be even. So, the orthogonality condition $\left(E(\phi),u_{0}^{\prime}\right)=0$ would be automatically satisfied.

Remark 3**.**

If the $\epsilon=0$ equation (4) has a unique even homoclinic solution $u_{0}$ (see [BL] for necessary and sufficient conditions on $f$), then the proof of persistence for $\epsilon$ small is considerably simplified by seeking $u_{\epsilon}=u_{0}+\phi$ with $\phi\in H^{2}(\mathbb{R})$ even. Note that given an even $g\in L^{2}(\mathbb{R})$, there exists a unique even $\phi\in H^{2}(\mathbb{R})$ such that $\phi^{\prime\prime}+f^{\prime}(u_{0})\phi=g$. Moreover $||\phi||_{H^{2}}\leq C||g||_{L^{2}}$ for some $C>0$ independent of $g$. This problem was briefly discussed at the end of [B].

2. Some remarks on the standard Lyapunov-Schmidt

approach

In this section we make some remarks on a difficulty that arises when trying to prove Theorem 1 using the standard Lyapunov-Schmidt reduction.

We have seen that $u_{0}$ satisfies (4) up to an order of $\epsilon$. We begin by refining this approximation so that $u_{ap}=u_{0}+\epsilon u_{1}$ satisfies (4) up to an order of $\epsilon^{2}$. We choose $u_{1}\in H^{2}(\mathbb{R})$, $u_{1}\bot u_{0}^{\prime}$ such that

[TABLE]

(this is possible via Lemma 1 and Proposition 1). Then, a simple calculation gives

[TABLE]

and thus from (12):

[TABLE]

We seek a solution of (4) in the form $u_{\epsilon}=u_{ap}+\psi_{\epsilon}$ with $\psi_{\epsilon}\in H^{2}(\mathbb{R})$. Then $\psi_{\epsilon}$ must satisfy

[TABLE]

where $L^{\epsilon}\psi=\psi^{\prime\prime}+f^{\prime}(u_{ap})\psi-\epsilon\Delta\psi$ and $N_{{ap}}(\psi)=-f(u_{ap}+\psi)+f(u_{ap})+f^{\prime}(u_{ap})\psi$.

Since $\Delta:H^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ is a bounded linear operator and $||u_{ap}-u_{0}||_{L^{\infty}}\leq C\epsilon$, $L^{\epsilon}$ is a regular $O(\epsilon)$ perturbation of $L^{0}$. From the special form of the perturbation, however, the simple eigenvalue [math] of $L^{0}$ is perturbed to an $O(\epsilon^{2})$ simple eigenvalue of $L^{\epsilon}$ (this is the source of the difficulty). We point out that such small eigenvalues would not have been present if we were in the symmetric setting of Remark 2. More precisely we have the following Proposition whose proof we postpone to Appendix B.

Proposition 2**.**

If $\epsilon\geq 0$ is sufficiently small then $\sigma(-L^{\epsilon})\subset\{\lambda_{1}(\epsilon)\}\cup[c,\infty)$ with $\lambda_{1}(\epsilon)$ simple corresponding to $\phi_{1}(\epsilon)\in H^{2}(\mathbb{R})$ with $||\phi_{1}(\epsilon)||_{H^{2}}=1$. Moreover $\lambda_{1}(\epsilon)$, $\phi_{1}(\epsilon)$ depend smoothly on $\epsilon$ up to $\epsilon=0$ and

[TABLE]

Define the orthogonal projection $P$ onto the span of $\phi_{1}$ by

[TABLE]

According to this projection we have

[TABLE]

where $X_{1},\ Y_{1}$ are respectively the kernel of $P$ in $H^{2}(\mathbb{R})$ and $L^{2}(\mathbb{R})$. By decomposing $\psi$ as $\psi=a\phi_{1}(\epsilon)+v$ $(a\in\mathbb{R},v\in X_{1})$, one finds that (17) is equivalent to

[TABLE]

Applying Proposition 2, using (16), and the Banach fixed point theorem, we can uniquely solve $(\ref{eq11})_{(i)}$ for $v=v^{*}(a,\epsilon)$ in a neighborhood of $(a,v)=(0,0)$. This solution depends smoothly on $a,\epsilon$ and satisfies $||v^{*}(a,\epsilon)||_{H^{2}}=O(a^{2}+\epsilon^{2})$ if $|a|,\epsilon\geq 0$ small. Using this in $(\ref{eq11})_{(ii)}$ and taking the $L^{2}$ inner product with $\phi_{1}(\epsilon)$ yields

[TABLE]

i.e.

[TABLE]

as $a,\epsilon\to 0$ with $\mu+\nu\geq 3$. If $\lambda_{1}(\epsilon)=d\epsilon+O(\epsilon^{2})$ with $d\neq 0$ (indep. of $\epsilon$), then we could apply the implicit function theorem to $\epsilon^{-2}B(\epsilon\tilde{a},\epsilon)=0$ and find an $a_{*}=O(\epsilon)$ satisfying (20). However, since by $(\ref{Lepsilon})_{(i)}$ we have $d=0$, this analysis breaks down.

Appendix A Proof of Lemma 1

$u\in L^{\infty}(\mathbb{R})$ and (H2) imply that $\Delta u\in L^{\infty}(\mathbb{R})$ and

[TABLE]

Then, by the Cauchy-Schwarz inequality, we get

[TABLE]

So,

[TABLE]

[TABLE]

[TABLE]

Thus, $\Delta u\in L^{2}(\mathbb{R}).$

We have

[TABLE]

[TABLE]

[TABLE]

Thus,

[TABLE]

Appendix B Proof of Proposition 2

Since zero is a simple eigenvalue of $-L^{0}$, it follows from regular perturbation theory (cf. Sec. 14.3 of [CH]) that it perturbs smoothly to a simple eigenvalue $\lambda(\epsilon)$ of $-L^{\epsilon}$. The corresponding eigenfunction $\phi(\epsilon)$ with $||\phi(\epsilon)||_{H^{2}}=1$ also depends smoothly (in the $H^{2}$ norm) on $\epsilon\geq 0$ small and $\phi(0)=\frac{u_{0}^{\prime}}{||u_{0}^{\prime}||_{H^{2}}}$. It is easy to show that $\lambda(\epsilon)$ is the principal eigenvalue of $-L^{\epsilon}$; we denote it by $\lambda_{1}(\epsilon)$ and the corresponding $H^{2}$ normalized eigenfunction by $\phi_{1}(\epsilon)$. (Recall that $(-L^{0}\phi,\phi)\geq c||\phi||_{L^{2}}^{2},\ \forall\phi\in H^{2}(\mathbb{R}),\ \phi\bot u_{0}^{\prime},$ and $\left|\left|\phi_{1}(\epsilon)-\frac{u_{0}^{\prime}}{||u_{0}^{\prime}||_{H^{2}}}\right|\right|_{H^{2}}\leq C\epsilon$, to obtain $(-L^{\epsilon}\phi,\phi)\geq c||\phi||_{L^{2}}^{2},\ \forall\phi\in H^{2}(\mathbb{R}),\ \phi\bot\phi_{1}(\epsilon)$.) We have

[TABLE]

and

[TABLE]

Since $\phi_{1}(\epsilon)\stackrel{{\scriptstyle H^{2}}}{{\to}}\frac{u_{0}^{\prime}}{||u_{0}^{\prime}||_{H^{2}}}$ as $\epsilon\to 0,$ we get

[TABLE]

Differentiating (15) yields

[TABLE]

i.e.

[TABLE]

This and the smoothness of $\lambda_{1}(\epsilon)$ gives us $(\ref{Lepsilon})_{(i)}$. From the smoothness of $\phi_{1}(\epsilon)$ (in the $H^{2}$ norm) we have $(\ref{Lepsilon})_{(ii)}$.

Acknowledgments. The author would like to thank the anonymous referees for carefully reading the paper and for offering several pertinent remarks. This work has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No 1889.