Asymptotic integrability and its consequences

TL;DR

This paper reviews asymptotic integrability in Hamiltonian systems, exploring its implications for soliton dynamics, quantization, and the origin of integrability in nonlinear wave physics.

Contribution

It introduces the concept of asymptotic integrability and links it to soliton theory, Hamiltonian dynamics, and the derivation of Lax pairs for integrable equations.

Findings

Asymptotic integrability leads to conserved integrals for wave packet propagation.

Exact asymptotic integrability implies the existence of Lax pairs for integrable equations.

Quantization in systems with asymptotic integrability reveals the origin of complete integrability.

Abstract

We give a brief review of the concept of asymptotic integrability, which means that the Hamilton equations for the propagation of short-wavelength packets along a smooth, large-scale background wave have an integral independent of the initial conditions. The existence of such an integral leads to a number of important consequences, which include, besides the direct application to the packets propagation problems, Hamiltonian theory of narrow solitons motion and generalized Bohr-Sommerfeld rule for parameters of solitons produced from an intensive initial pulse. We show that in the case of systems with two wave variables and exact fulfillment of the asymptotic integrability condition, the `quantization' of mechanical systems, associated with the additional integrals, yields the Lax pairs for a number of typical completely integrable equations, and this sheds new light on the origin of the complete integrability in nonlinear wave physics.