Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations I

Abstract

Explicit formulae involving determinants are obtained for the solutions of a class of linear differential-difference and difference-difference evolution equations. The corresponding non-linear problems generated by the conditions of compatibility of those linear equations (the discrete analogues of the Zakharov-Schabat equations) will be discussed in a forthcoming paper. The general idea of this paper is in close analogy with the approach used in the previous works of the author [1, 2] based on the property of the Darboux invariance of the associated linear problem. Surprisingly, for the difference equations, most of the formulae and their derivation are even simpler than the continuous case considered in [1, 2] (see also the works [3–5].