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

Abstract

We construct a solution of an analog of the Schrödinger 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 Schrödinger 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.

arXiv comment13 pages, in Russian. There is an English translation of the text containing misprints in the formula (5), (19) and in the first equation of the concluding observations

References in corpus (6)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• Quantum Painleve-Calogero Correspondence
• The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation
• Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations
• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation
• Asymptotics for a special solution to the second member of the Painleve I hierarchy

Cited by in corpus (1)

• On the tritronquée solutions of P${}_I^2$