Quasiclassical integrability condition in AKNS scheme

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 data. If this condition is fulfilled, then the carrier wave number of any packet is a certain function of the local values of the dispersionless variables pertained to the soliton equation under consideration. We show by 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.