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Zbl 0682.34011
Gibbon, J.D.; Newell, A.C.; Tabors, M.; Zeng, Y.B.
Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations. (English)
[J] Nonlinearity 1, No.3, 481-490 (1988). ISSN 0951-7715; ISSN 1361-6544/e

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 2000:: *34A34 Nonlinear ODE and systems, general
35Q99 PDE of mathematical physics and other areas

Keywords: second Painlevé transcendent; Korteweg-de Vries equation; travelling wave reduction; special solutions; ordinary differential equations; Weiss-Tabor-Carnevale procedure; Lax pairs; Bäcklund transformations; fourth Painlevé transcendents

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