15 - Inverse Scattering for the Matrix Schrö Equation with Non-Hermitian Potential

Summary

INTRODUCTION.

Kamijo and Wadati (1974) developed an inverse scattering formalism for the matrix Schrödinger equation with Hermitian potential matrix U(x). Assuming a similar formalism in the non-Hermitian case the authors and later on Calogero and Degasperis (1976) found several classes of solvable nonlinear evolution equations. To our knowledge this assumption has nowhere been justified in the literature. Indeed, Wadati (1980) writes: “At the moment, it is an open question as to what are the most general conditions for which the inverse scattering problem can be solved. I have considered three cases:

The first crucial point in the discussions is the definition of the Wronskian. Although we need extra assumptions, similar arguments seem to be valid”.

The present paper is an attempt to settle this question. We consider the Schrödinger scattering problem for any continuous potential matrix U(x) decaying sufficiently rapidly for continuous potential matrix U(x) decaying sufficiently rapidly for. Superimposing some rather natural regularity conditions on the right transmission coefficient that are partly familiar from the Zakharov-Shabat scattering problem (Eckhaus and van Harten, 1981), we show that - apart from the unique solvability of the Gel'fand-Levitan-Marchenko equation - the inverse scattering problem can be solved completely. Though initially the approach parallels that of Kamijo and Wadati (1974), our ways split up in Section 5. The reason is that the Wronskians employed by Kamijo and Wadati to extend the scattering coefficients to the upper half plane are of no use in the non-Hermitian case. To circumvent this difficulty, we perform the extension by means of integral representations in terms of the potential and the Jost functions.

NOTATION.

Throughout, we shall use the following notation:

The determinant of A will be denoted by det A. I is the n x n identity matrix. We set

Our notation for the Jost functions, scattering coefficients, etc. closely resembles that used in Eckhaus and van Harten (1981). This notation is different from the one in Kamijo and Wadati (1974). For convenience, we specify here the relation between these different notations: