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The Painlevé property and Hirota’s method. (English) Zbl 0581.35074

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota’s method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self-truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota’s method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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