Cnoidal Waves on Fermi-Pasta-Ulam Lattices

TL;DR

This paper demonstrates that Fermi-Pasta-Ulam lattices support exact periodic travelling waves resembling KdV cnoidal waves, which maintain their shape indefinitely and exhibit properties suggesting long-term stability and energy transport.

Contribution

It establishes the existence of exact, long-time stable, periodic travelling waves in FPU lattices that asymptotically resemble KdV cnoidal waves, extending previous formal asymptotic results.

Findings

Exact periodic travelling waves are stable over infinite time.

These waves carry nonzero energy per particle.

Behavior suggests long-term stability under nonlinear interactions.

Abstract

We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schr\"odinger operator with KdV cnoidal wave potential.