Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices

TL;DR

This paper demonstrates the existence of nanopteron traveling waves in a spring dimer FPUT lattice, revealing complex wave profiles combining localized solitons and small periodic oscillations, extending previous diatomic lattice results.

Contribution

It generalizes the analysis of traveling waves to spring dimers with complex nonlinearities, introducing new methods to handle superposition of different function types and confirming wave behaviors specific to spring dimers.

Findings

Existence of nanopteron traveling waves in spring dimer FPUT lattices.

Wave profiles are superpositions of solitons and small periodic terms.

Spring dimer waves exhibit alternating behavior between particle sites.

Abstract

We study the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. This is a one-dimensional lattice of identical particles connected by alternating nonlinear springs. Following the work of Faver and Wright on the mass dimer, or diatomic, lattice, we find that the lattice equations in the long wave regime are singularly perturbed and apply a method of Beale to produce nanopteron traveling waves with wave speed slightly greater than the lattice's speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV sech2-type soliton) and a periodic term of very small amplitude. Generalizing our work in the diatomic case, we allow the nonlinearity in the spring forces to have the more complicated form "quadratic plus higher order terms." This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of function types from Beale's ansatz. Unlike the diatomic case, the value of the leading order term in the traveling wave profiles alternates between particle sites, so that the spring dimer traveling waves are also "stegotons," in the terminology of LeVeque and Yong. This behavior is absent in the mass dimer and confirms the approximation results of Gaison, Moskow, Wright, and Zhang for the spring dimer.