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Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations. (English) Zbl 0682.34011
Summary: We investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Bäcklund transformations for systems of ordinary differential equations. This method can yield both auto-Bäcklund transformations, and where necessary, Bäcklund transformations between different equations. In the latter case we investigate the circumstances under which the general Bäcklund transformations reduce to auto-Bäcklunds. In addition, special solution families for the second and fourth Painlevé transcendents are obtained.
MSC:
34A34Nonlinear ODE and systems, general
35Q99PDE of mathematical physics and other areas