Spectral transform for nonlinear evolution equations with N space dimensions

Abstract

A method generalizing direct linearization methods to get solutions of nonlinear systems was recently visited by the author. His aim was to show how the classical results obtained in problems with two space dimensions (and time) could be extended to N space dimensions, N > 2. In particular, he showed that this can still be done with solutions written as 2×2 matrices, and with more than two spectral variables, but that the differential operators which appear inside each of the coupled equations can always be expressed in terms of two independent variables only. This restriction, however, does not prevent the existence of genuine three-dimensional solitons inside solutions of the coupled systems. In the present Letter, the method is derived with solutions written as N× N matrices, and two spectral variables only. Linear restrictions disappear, but nonlinear constraints come in. The evolution equations for the N× N matrix-valued solutions are related to N first order and one second order linear differential operators, which together define the “spectral problem”. These operators commute, as generalized Lax pairs, on the null-space common to the first order operators. Writing down the commutation conditions both yields evolution equations for the off-diagonal elements of the solution and equations relating some of their spatial derivatives to products of two other elements. Hence the solutions we construct are restricted to manifolds of the N-dimensional space.