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Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. I. Kaup-Kupershmidt equation. (English) Zbl 0932.35180

Summary: Among the 50 nonlinear second-order differential equations of Painlevé and Gambier, those which are linearizable provide a natural scheme for deriving the Lax pair and the Darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three. This new method allows one to obtain, from singularity analysis only, the Bäcklund transformation of the Kaup-Kupershmidt equation, a result which was missing until now.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35A30 Geometric theory, characteristics, transformations in context of PDEs
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