“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Abstract

We consider two compatible linear evolution equations with times $s_1$ and $s_2$ depending on two spatial variables. These evolution equations are the analogues of the non-stationary Schrödinger equations determined by the two Hamiltonians $H^{\frac{7}{2}+1}_{s_k}(s_1,s_2, q_1,q_2, p_1, p_2)$ $(k=1,2)$ of the Hamilton system $H^{\frac{7}{2}+1}$ formed by a pair of compatible Hamiltonian systems of equations admitting the application of isomonodromic deformations method. These analogues arise from canonical non-stationary Schrödinger equations determined by the Hamiltonians $H^{\frac{7}{2}+1}_{s_k}$. They arise by the formal replacement of the Planck constant by the imaginary unit. We construct explicit solutions of these analogues of Schrödinger equations in terms of the solutions of the corresponding linear systems of ordinary differential equations in the isomonodromic deformations method, whose compatibility condition is the Hamiltonian system $H^{\frac{7}{2}+1}$. The key role in the construction of these explicit solutions is played by the change, which was used earlier in constructing the solutions of non-stationary Schrödinger equation determined by the Hamiltonians of isomonodromic Hamiltonian Garnier system with two degrees of freedom as well as of two isomonodromic degenerations of the latter. We discuss the applicability of this change for constructing the solutions to analogues of non-stationary Schrödinger equations determined by the Hamiltonians of the entire hierarchy of isomonodromic Hamiltonian systems with two degrees of freedom being the degenerations of this Garnier system. We mention also a relation of solutions to Hamilton systems $H^{\frac{7}{2}+1}$ with some problems of modern nonlinear mathematical physics. In particular, we show that the solutions of these Hamiltonian systems are determined explicitly by the simultaneous solutions to the Korteweg-de Vries equation $u_t+u_{xxx}+uu_x=0$ and a non-autonomous fifth order ordinary differential equations, which are used in universal description of the influence of a small dispersion on the transformation of weak hydrodynamical discontinuities into the strong ones.