Counterpropagating Two-Soliton Solutions in the FPU Lattice

TL;DR

This paper proves the global existence and stability of counterpropagating two-soliton solutions in the FPU lattice, showing they behave like superpositions of solitary waves and remain stable under perturbations.

Contribution

It establishes the first rigorous proof of stable, interacting two-soliton solutions in the FPU lattice with general nearest-neighbor potentials.

Findings

Solutions are close to superpositions of two solitary waves for large times.

Counterpropagating solutions are stable under $ ext{l}^2$ perturbations.

Solutions are asymptotically stable with exponentially decaying perturbations.

Abstract

We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $\ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $\pm \infty$.}