Asymptotic integrability of nonlinear wave equations

TL;DR

This paper introduces the concept of asymptotic integrability in nonlinear wave equations, linking the preservation of Hamiltonian structure during hydrodynamic evolution to the existence of an additional integral of motion.

Contribution

It formalizes asymptotic integrability, connecting it to the quasiclassical limit of Lax pairs and providing illustrative examples.

Findings

Asymptotic integrability condition expressed as a system of equations.

Relation between solutions and quasiclassical limits of Lax pairs.

Examples demonstrating the application of the theory.

Abstract

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave, so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related with the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory by several examples.