The Concept of monodromy for linear problems and its applications

Abstract

The notion of a monodromy matrix (operator) naturally appears under the study of linear systems with periodic coefficients. This notion gives rise to the well known result [4, 17] on the reductibility of linear periodic systems (Floquet's theorem) wich says that the monodromy matrix contains the complete information about a given system. The goal of the present work is to develop an unified viewpoint of monodromy for linear systems on Lie algebras with quasi periodic and decreasing boundary conditions. The quasi-periodic case is a natural generalization of the periodic case, and the decreasing case can be interpreted as the “limiting” periodic case (the period tends to infinity). Such a class of linear problems arise in the integrability theory of nonlinear partial differential equations in the framework of the so-called inverse scattering method [8, 11] If a nonlinear partial differential equations can be represented as the consistence condition for two linear problems (called the L-A pair). Then the inverse scattering method allow us reconstruct a wide class of solutions of the nonlinear equations from the corresponding “spectral data”. This leads to the study of the zero curvature equation and Lax equation [8, 11]. The main point here is the study of the analytic properties of the monodromy matrix depending on a spectral parameter and then the time-evolution of the spectral data. As an application of the general concept of monodromy, we consider a class of linear problems associated with integrability of nonlinear schödinger equation (NLS equation) [8, 11].