Non-isospectral and isomonodromic deformations: self-similar solutions

Abstract

We construct self-similar solutions of various soliton equations obtained with the help of the inverse scattering transform with a variable spectral parameter. We demonstrate that corresponding self-similar systems, which represent nonlinear ordinary differential equations (ODEs), may be divided into two classes. The first class contains equations that can be directly solved in the framework of the method of isomonodromic deformations. Some of the equations may be regularly reduced to certain Painleve equations. Equations in the second class include variable coefficients that satisfy additional nonlinear ODEs. We prove that at least one such additional ODE for the coefficients is without the Painleve property. As far as we know, there is no regular method that can be used to solve these supplementary equations. The second class contains really new equations that one could solve, in principle, by the method of isomonodromic deformations only after one finds a solution to the additional ODE. The most interesting ODE. From the physical point of view, is a self-similar reduction for the Maxwell-Bloch system with pumping, which has applications to nonlinear optics.