Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations |
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| Authors: | Deng‐Shan Wang Jiang Liu Zhifei Zhang |
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| Affiliation: | 1. School of Applied Science, Beijing Information Science and Technology University, Beijing, China;2. Department of Systems Science, Business School, University of Shanghai for Science and Technology, Shanghai, China;3. Department of Mathematics, Huazhong University of Science and Technology, Wuhan, China |
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| Abstract: | In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd. |
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| Keywords: | coupled Korteweg– de Vries equation integrability invertible transformation Lax pair Painlevé integrability exact solution |
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