Spectral transform for nonlinear evolutions in N-dimensional spaces

Abstract

A method which generalizes the dressing method, the delta -method and the direct linearizing method is presented, and the conditions for which it yields solutions of nonlinear systems with N space variables are derived. The method is illustrated for 2*2 matrix-valued functions phi . phi depends on the space and time variables and the main spectral variable k, and is defined as the solution of an integral equation. The method depends on eight parameters instead of four generally used or five used recently. When a linear differential operator 'compatible' with the dispersion relation is applied to phi , it yields an equation for phi , which involves this function and its asymptotic values as k goes to infinity. three equations of this type can be combined to construct a closed system of nonlinear equations for the elements of the leading asymptotic value Q of k( phi -1). When the eight parameters correspond to two independent 'reflection coefficients', the two lines of phi can be treated separately (two-channels problem). If symmetry relations and suitable transformation of variables reduce the problem to one reflection coefficient only, the two lines are coupled. In both cases, there are only three possible final results.