Long-Time approximations of small-amplitude, long-wavelength FPUT solutions

TL;DR

This paper derives explicit long-time approximation results for small-amplitude, long-wavelength FPUT solutions using the defocusing mKdV equation as a modulation equation, including non-localized solutions like kinks, and establishes their meta-stability.

Contribution

It introduces new approximation estimates for FPUT solutions via the mKdV equation, accommodating non-localized kink solutions and analyzing their meta-stability.

Findings

Explicit approximation estimates for FPUT using mKdV

Inclusion of non-localized kink solutions in approximations

Meta-stability results for kink-like FPUT solutions

Abstract

It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other such results have been proved in this case. However, situations in which the defocusing modified KdV (mKdV) equation is expected to be the modulation equation have been much less studied. As seen in numerical experiments, the kink solution of the mKdV seems essential in understanding the $\beta$-FPUT recurrence. In this paper, we derive explicit approximation results for solutions of the FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for solutions to be non-localized as to allow approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT.