The solution to the q-KdV equation

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this Letter is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, first one by Frenkel [Int. Math. Res. Notices 2 (1996) 55] and a variation by Khesin, Lyubashenko and Roger [J. Func. Anal. 143 (1997) 55]. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where Df(x) = f(qx). Therefore every notion about the 1-Toda lattice can be transcribed into q-language, (C) 1998 Elsevier Science B.V.