Non-autonomous H\'enon-Heiles Systems
Abstract
Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable H\'enon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited B\"acklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlev\'e transcendent.
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