Self-Similar Isomonodromy Solutions of Nonlinear Schrödinger Equation

Abstract

Recent works of a group of Japanese mathematicians M. Sato, T. Miwa, M. Jimbo, K. Ueno, E. Date and others [1–2] have attracted major attention of researchers in various areas of mathematics and mathematical physics. One of the interesting gains of the theory of nonlinear equations of mathematical physics is the concept of “isomonodromy” solutions of nonlinear equations admitting a Lax representation. Isomonodromy solutions are a natural generalization of their self similar solutions. Within the framework of the formalism of the matrix Riemann-Hilbert problem finally adopted, isomonodromy solutions are characterized by the constancy in the “spectral parameter” λ of all junction matrices. This leads to the situation that the function corresponding to an isomonodromy solution satisfies, together with the original U – V system, also a third equation in λ. The data of the Riemann-Hilbert problem here acquire the meaning of monodromy data for the λ-equation. The apriori independence of the monodromy data from the dynamical variables x and t means that the original nonlinear equation describes isomonodromy deformations of the coefficients of the λ-equation. This is the motivation for the term isomonodromy solution.