Moving modulating pulse and front solutions of permanent form in a FPU model with nearest and next-to-nearest neighbor interaction

TL;DR

This paper proves the existence of moving modulating front and pulse solutions in a generalized FPU lattice with both nearest and next-to-nearest neighbor interactions, using spatial dynamics and center manifold techniques.

Contribution

It introduces the first rigorous proof of moving modulating solutions in a generalized FPU model with complex interactions, extending classical results.

Findings

Existence of moving modulating front solutions with small tails

Existence of localized pulse solutions when potentials are even

Solutions approximated by nonlinear Schrödinger solitary waves

Abstract

We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical FPU lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of nearest neighbor attraction and next-to-nearest neighbor repulsion we prove that such a lattice admits, in contrast to the classical FPU model, moving modulating front solutions of permanent form, which have small converging tails at infinity and can be approximated by solitary wave solutions of the Nonlinear Schr\"odinger equation. When the associated potentials are even, then the proof yields moving modulating pulse solutions of permanent form, whose profiles are spatially localized. Our analysis employs the spatial dynamics approach as developed by Iooss and Kirchg\"assner. The relevant solutions are constructed on a five-dimensional center manifold and their persistence is guaranteed by reversibility arguments.