"Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom

Details "Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom "Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom

2012-04-18

arxiv.org/abs/1204.4006v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

13 pages, in Russian

Scientific paper

We construct a solution of an analog of the Schr\"{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of variables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schr\"{o}dinger equation corresponding to the Hamiltonian $ H_{II} (z, t, q_1, q_2, p_1, p_2) $ of Hamiltonian system with respect to $t$ which is compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy.

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