Lax Pairs: Integrable, Less Integrable and Nonintegrable Systems

TL;DR

This paper reviews the spectrum of behaviors in systems with Lax Pair formulations, from integrable to nonintegrable, highlighting differences in solution regularity and recent theoretical connections.

Contribution

It provides a comparative review of integrable, less integrable, and nonintegrable systems with Lax Pairs, including new results on boundary value problems and perturbed equations.

Findings

Integrable systems exhibit tame, predictable behavior.

Some systems with Lax Pairs show irregular, chaotic dynamics.

Connections are made between perturbed Lax Pair equations and existing theory.

Abstract

Completely integrable finite dimensional Hamiltonian systems are well understood thanks to the work of Liouville and Arnold. On the other hand, the Lax Pair formulation of the KdV equation marks the beginning of the extension of the completely integrable theory to infinite dimensional Hamiltonian systems. Solutions of initial value problems for systems that admit a Lax Pair formulation normally have a tame qualitative behavior if Lax Pairs give rise to an infinite complete set of conserved laws. The situation is different for initial-boundary value problems, even in one space dimension. There are problems where integrability persists and regular (long time asymptotic) behavior can be proven (and we have proven it). There are others where even irregular "fractal-chaotic-looking" behavior can appear. In this short article we review an instance of each case. We also make a connection with results from the existing theory of perturbed Lax Pair equations on the real line.