Self-similar solutions of equations of the nonlinear Schrodinger type

A study is made of self-similar solutions of an entire family of one-dimens ional integrable dynamic systems of the nonlinear Schrodinger equation type . This family is reduced to one of three canonical forms corresponding to a Toda chain, a Volterra chain, or to the Landau-Lifshitz model, which can a lso be reduced to three self-similar systems coupled by Miura transformatio ns with the fourth Painleve equation. A commutative representation is const ructed for this equation. A relationship is established between the poles o f the rational solutions of the fourth Painleve equation and the steady-sta te distribution of the electric charges in a parabolic potential. A self-si milar solution is constructed for the spin dynamics. An exact solution is o btained for the nonlinear Schrodinger equation with variable dispersion (op tical soliton). (C) 2000 MAIK "Nauka/Interperiodica".