The solution to the q-KdV equation |
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| Institution: | 1. Department of Mathematics, Brandeis University, Waltham, MA 02254, USA;2. Department of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., Sofia 1126, Bulgaria;3. Université de Louvain, 1348 Louvain-la-Neuve, Belgium;1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;2. Department of Basic, Shijiazhuang Post and Telecommunication Technical College, Shijiazhuang 050021, China;1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia;2. Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4a, Moscow, 125047, Russia;3. National Research Nuclear University MEPHI, Kashirskoye sh. 31, Moscow, 115409, Russia |
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| Abstract: | Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this Letter is to show that and 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 D f(x) = f(qx). Therefore every notion about the 1-Toda lattice can be transcribed into q-language. |
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