Homoclinics, rogue waves and breathers in nonlinear lattice wave equations

Abstract

We prove the existence of rogue waves and breathers in nonlinear lattice wave equations including the nonlinear Klein--Gordon and the FPUT lattice (with added local forces). The main feature of our lattice wave equation is that the kinetic part $\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}$ is multiplied by a non-constant function $\frac{1}{d(t)}$ such that gaps in the spectrum of $\frac{1}{d(t)}\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}$ open wide enough to include the spectrum of the spatial linear operator. We find solutions as critical points of an indefinite functional using a saddle-point method combined with concentration-compactness arguments. In case of temporally $T$-periodic coefficients and under identical assumptions as for rogue waves, the same variational method also provides existence of breather solutions whose temporal period is an arbitrary prescribed integer multiple of $T$.