Darboux transforms, deep reductions and solitons

Abstract

Explicit formalisms for deep reductions of matrix differential equations and Darboux covariance properties are presented. The matrix 2*2 spectral problem of the second order is considered. This problem with the reduction constraints being imposed on the potentials is the first equation of the Lax representation of the Hirota-Satsuma system (1981). The two reduction cases are treated with the help of the bilinear delta -forms. The covariance of these forms with respect to the Darboux transforms under restrictions gives rise to explicit formulas of N-soliton solutions. In particular the two-parameter soliton solutions of the Hirota-Satsuma system are obtained. The specific feature of such solutions evolution is that the singularity appears in some parameter region. The Yajima-Oikawa system (1976) is given as an example of the technique application to a 3*3 spectral problem.