Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity

TL;DR

This paper proves the existence of special solitary wave solutions, called nanopterons, in a diatomic FPUT lattice, revealing complex wave interactions due to optical band effects not seen in monatomic lattices.

Contribution

It demonstrates the existence of nanopteron solutions in diatomic FPUT lattices, highlighting the singular perturbation nature and optical band wave effects.

Findings

Existence of nanopteron solutions near the lattice's sound speed

Presence of optical band waves with arbitrary phase speed

Contrast with monatomic FPUT behavior

Abstract

We study the existence of solitary waves in a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. For monatomic FPUT the traveling wave equations are a regular perturbation of the Korteweg-de Vries (KdV) equation's but, surprisingly, we find that for the diatomic lattice the traveling wave equations are a singular perturbation of KdV's. Using a method first developed by Beale to study traveling solutions for capillary-gravity waves we demonstrate that for wave speeds in slight excess of the lattice's speed of sound there exists nontrivial traveling wave solutions which are the superposition an exponentially localized solitary wave and a periodic wave whose amplitude is extremely small. That is to say, we construct nanopteron solutions. The presence of the periodic wave is an essential part of the analysis and is connected to the fact that linear diatomic lattices have optical band waves with any possible phase speed.