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The solution to the q-KdV equation

M. Adler·E. Horozov·P. van Moerbeke
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Abstract

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper 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, one by Frenkel and a variation by Khesin et al. 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.

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M. Adler, E. Horozov, P. van Moerbeke. 1997-12-19. The solution to the q-KdV equation. https://doi.org/10.1016/s0375-9601(98)00082-6

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