On Soliton Resolution for a Lattice

TL;DR

This paper explores the extension of the soliton resolution conjecture from PDEs to discrete Fermi-Pasta-Ulam-Tsingou lattices, revealing complex phenomena including solitons, periodicity, and modulated oscillations through numerical analysis.

Contribution

It investigates the soliton resolution conjecture in discrete lattices, identifying a richer set of phenomena beyond solitons, including modulated oscillations, especially in non-integrable and chaotic cases.

Findings

Pure periodicity, solitons, and modulated oscillations observed in numerical simulations.

The phenomena persist in small perturbations of integrable lattices.

Chaotic behaviors can also occur beyond integrable cases.

Abstract

The soliton resolution conjecture for evolution PDEs of dispersive type states (vaguely) that generic initial data of finite energy give rise asymptotically to a set of receding solitons and a decaying background radiation. In this letter, we investigate a possible extension of this conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather than PDEs) in two cases: the case of finite energy initial data and a more general case where the initial data are a short range perturbation of a periodic function. In the second case, inspired by rigorous results on the Toda lattice, we suggest that the soliton resolution phenomenon is replaced by something somewhat more complicated: a short range perturbation of a periodic function actually gives rise to different phenomena in different regions. Apart from regions of (asymptotically) pure periodicity and regions of solitons in a periodic background, we also observe "modulated" regions of fast oscillations with slowly varying parameters like amplitude and phase. We have conducted some numerical calculations to investigate if this trichotomy (pure periodicity + solitons + modulated oscillations) persists for any discrete lattices of the Fermi-Pasta-Ulam-Tsingou type. For small perturbations of integrable lattices like the linear harmonic lattice, the Langmuir chain and the Toda lattice, this is true. But in general even chaotic phenomena can occur.