Breather Solutions for a Quasilinear $(1+1)$-dimensional Wave Equation

TL;DR

This paper constructs time-periodic, spatially localized breather solutions for a specific quasilinear wave equation related to electromagnetic waves in Kerr media, demonstrating their exponential decay and approximation by Fourier truncation.

Contribution

It introduces a novel approach to find breather solutions in a quasilinear wave equation using Fourier series and variational methods, with explicit examples and infinitely many solutions.

Findings

Existence of exponentially localized breather solutions.

Solutions can be approximated by finite Fourier series.

Infinitely many solutions in explicit step potential cases.

Abstract

We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with $g\not\equiv 0$ and $h(x)=\gamma\,\delta_0(x)$ with $\gamma\in{\mathbb{R}}\backslash\{0\}$ and $\delta_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-\frac{d^2}{dx^2}-k^2\omega^2g$ on $L^2(\mathbb{R})$ for all $k\in 2\mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitly given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.