Breather solutions for a quasi‐linear ( 1 + 1 ) $(1+1)$ ‐dimensional wave equation

Abstract We consider the( 1 + 1 ) $(1+1)$ ‐dimensional quasi‐linear wave equationg ( x ) w t t − w x x + h ( x )( w t 3 ) t = 0 $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ onR × R $\mathbb {R}\times \mathbb {R}$ that arises in the study of localized electromagnetic waves modeled by Kerr‐nonlinear Maxwell equations. We are interested in time‐periodic, spatially localized solutions. Hereg ∈ L ∞ ( R )$g\in {L^{\infty }(\mathbb {R})}$ is even withg ≢ 0 $g\not\equiv 0$ andh ( x ) = γδ 0 ( x ) $h(x)=\gamma \,\delta _0(x)$ withγ ∈ R ∖ { 0 } $\gamma \in \mathbb {R}\backslash \lbrace 0\rbrace$ andδ 0 $\delta _0$ the delta‐distribution supported in 0. We assume that 0 lies in a spectral gap of the operatorsL k = − d 2 d x 2− k 2 ω 2 g $L_k=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-k^2\omega ^2g$ onL 2 ( R ) ${L^{2}(\mathbb {R})}$ for allk ∈ 2 Z + 1 $k\in 2\mathbb {Z}+1$ together with additional properties of the fundamental set of solutions ofL k $L_k$ . By expanding w $w$ 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 $g$ . In these examples, we even find infinitely many distinct breathers.