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- W1999331503 abstract "The six Painlevé equations can be written in the Hamiltonian form, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of the Painlevé equations as symplectic birational transformations in C4 and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. We also show that the original Painlevé equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions." @default.
- W1999331503 created "2016-06-24" @default.
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- W1999331503 date "2015-02-01" @default.
- W1999331503 modified "2023-10-17" @default.
- W1999331503 title "Painlevé equations, elliptic integrals and elementary functions" @default.
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- W1999331503 doi "https://doi.org/10.1016/j.jde.2014.10.018" @default.
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