Approximation of (some) FPUT lattices by KdV Equations

TL;DR

This paper proves that under certain random conditions, small amplitude, long wavelength solutions of FPUT lattices are almost surely well-approximated by KdV equations over long times, using energy estimates, homogenization, and autoregressive processes.

Contribution

It introduces a simple condition on randomness that ensures rigorous approximation of FPUT lattices by KdV equations, combining energy estimates with homogenization techniques.

Findings

Almost sure approximation of FPUT by KdV under specific randomness conditions

Long-time validity of the approximation for small amplitude solutions

Novel application of autoregressive processes in the proof

Abstract

We consider a Fermi-Pasta-Ulam-Tsingou lattice with randomly varying coefficients. We discover a relatively simple condition which when placed on the nature of the randomness allows us to prove that small amplitude/long wavelength solutions are almost surely rigorously approximated by solutions of Korteweg-de Vries equations for very long times. The key ideas combine energy estimates with homogenization theory and the technical proof requires a novel application of autoregressive processes.