Gibbon, J. D.; Newell, A. C.; Tabors, M.; Zeng, Y. B. Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations. (English) Zbl 0682.34011 Nonlinearity 1, No. 3, 481-490 (1988). 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. Cited in 11 Documents MSC: 34A34 Nonlinear ordinary differential equations and systems 35Q99 Partial differential equations of mathematical physics and other areas of application 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 PDFBibTeX XMLCite \textit{J. D. Gibbon} et al., Nonlinearity 1, No. 3, 481--490 (1988; Zbl 0682.34011) Full Text: DOI