# Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio

## TL;DR

This paper proves the existence of nanopteron solutions in diatomic FPUT lattices with small mass ratios, revealing complex wave behaviors that are not true solitary waves and depend on specific mass ratios.

## Contribution

It establishes the existence of nanopteron traveling waves in diatomic FPUT lattices as the mass ratio approaches zero, addressing a singular limit and analyzing associated nonlocal Schrödinger operators.

## Key findings

- Traveling waves are nanopterons, not true solitary waves.
- Solutions exist only for certain open sets of small mass ratios.
- Wave solutions asymptote to small amplitude periodic waves at infinity.

## Abstract

Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a Fermi-Pasta-Ulam-Tsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather "nanopterons", which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schr\"odinger operator in its semi-classical limit.