On asymptotic solutions of integrable wave equations

Asymptotic 'soliton train' solutions of integrable wave equations described by inverse scattering transform method with second-order scalar eigenvalue problem are considered. It is shown that if asymptotic solution can be pre sented as a modulated one-phase nonlinear periodic wavetrain, then the corr esponding Baker-Akhiezer function transforms into quasiclassical eigenfunct ion of the linear spectral problem in weak dispersion limit for initially s mooth pulses. In this quasiclassical limit the corresponding eigenvalues ca n be calculated with the use of the Bohr Sommerfeld quantization rule. The asymptotic distributions of solitons parameters obtained in this way specif y the solution of the Whitham equations. (C) 2001 Elsevier Science B.V. All rights reserved.