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A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

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  • Ma, Wen-Xiu
  • Lee, Jyh-Hao

Abstract

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo–Miwa equation is treated, together with a Bäcklund transformation.

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  • Ma, Wen-Xiu & Lee, Jyh-Hao, 2009. "A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1356-1363.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:3:p:1356-1363
    DOI: 10.1016/j.chaos.2009.03.043
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    1. He, Ji-Huan & Wu, Xu-Hong, 2006. "Exp-function method for nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 700-708.
    2. Chen, Yong & Yan, Zhenya, 2005. "New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 399-406.
    3. Abdou, M.A., 2007. "The extended F-expansion method and its application for a class of nonlinear evolution equations," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 95-104.
    4. Wang, Dengshan & Zhang, Hong-Qing, 2005. "Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 601-610.
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