Quasiclassical integrability condition in AKNS scheme

TL;DR

This paper investigates the quasiclassical integrability condition in soliton equations within the AKNS scheme, revealing how the Hamiltonian structure is preserved in the dispersionless limit and determining the quasiclassical Lax pair functions.

Contribution

It introduces a specific integrability condition for soliton equations in the AKNS scheme and demonstrates its implications for the quasiclassical limit of Lax pairs.

Findings

The Hamiltonian structure is preserved by dispersionless flow under the integrability condition.

Carrier wave number functions relate to local dispersionless variables and determine quasiclassical Lax pairs.

Examples show the connection between wave number functions, dispersion relations, and Lax pair limits.

Abstract

In this paper, we study the condition of quasiclassical integrability of soliton equations. This condition states that the Hamiltonian structure of equations, which govern propagation of high-frequency wave packets, is preserved by the dispersionless flow independently of initial conditions. If this condition is fulfilled, then the carrier wave number of any packet is a certain function of local values of the dispersionless variables pertained to the soliton equations under consideration. We show for several examples that this function together with the dispersion relation for linear harmonic waves determine the quasiclassical limit of the Lax pair functions in the scalar representation of the Ablowitz-Kaup-Newell-Segur scheme.