The solution to the q-KdV equation
(1998) Physics Letters. Section A: General, Atomic and Solid State Physics — Vol. 242, n° 3, p. 139-151 (1998)
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- 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.
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Adler, M., Horozov, E., & Van Moerbeke, P. (1998). The solution to the q-KdV equation. Physics Letters. Section A: General, Atomic and Solid State Physics, 242(3), 139-151. https://doi.org/10.1016/S0375-9601(98)00082-6 (Original work published 1998)