We propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painlevé property, a Bäcklund transformation is defined in terms of an expansion about the singular manifold. This Bäcklund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati‐type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Bäcklund transformations of the modified equations provides a method for the iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev–Petviashvili equation and the Hirota–Satsuma equations.
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September 1985
Research Article|
September 01 1985
Modified equations, rational solutions, and the Painlevé property for the Kadomtsev–Petviashvili and Hirota–Satsuma equations
John Weiss
John Weiss
Center for Studies of Nonlinear Dynamics, La Jolla Institute, 8950 Villa La Jolla Drive, Suite 2150, La Jolla, California 92037 and Institute for Pure and Applied Physical Science, University of California, San Diego, La Jolla, California 92093
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J. Math. Phys. 26, 2174–2180 (1985)
Article history
Received:
October 31 1984
Accepted:
February 22 1985
Citation
John Weiss; Modified equations, rational solutions, and the Painlevé property for the Kadomtsev–Petviashvili and Hirota–Satsuma equations. J. Math. Phys. 1 September 1985; 26 (9): 2174–2180. https://doi.org/10.1063/1.526841
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