Non-autonomous H\'{e}non-Heiles Systems

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\'{e}non-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\"{a}cklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlev\'{e} transcendent.