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

Abstract

Abstract. We consider two compatible linear evolution equations with times 1 and 2 depending on two spatial variables. These evolution equations are the analogues of the non-stationary Schrödinger equations determined by the two Hamiltonians 7 2 +1 ( 1 , 2 , 1 , 2 , 1 , 2 ) ( = 1, 2) of the Hamilton system +1 . 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 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 +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 + + = 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.

Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$

“…Novikov observed a similarity of this change with formula (2.3.36) in [23]. For system 2+1+1+1 , this justifies an assumption of paper [32] that this change can be useful in constructing analogues of non-stationary Schrödinger equations defined by the Hamiltonians of all degenerations of Garnier system. Apart of such change, it is useful to have in mind the changes being quantum analogues of known classical transformations.…”

“…In constructions of the present paper, change (17) plays an important role. Earlier, the same change was successfully applied in papers [31], [32] and [36], in which there were constructed solutions to analogues of non-stationary Schrödinger equations defined by the Hamiltonians of Garnier system as well as by some of its degenerations. Earlier, for other purposes, this change was employed by D.P.…”

“…In work [31], there was formulated an opinion that after a generalization of a known procedure of step-by-step degenerations of the terms in Garnier hierarchy on the quantum level [9], [10], by the results in [31] one should get immediately compatible solutions to pair of equations (4) for all representative of this hierarchy. However, as it was noted in [32], in this way there are troubles related to the fact that in some of step-by-step degenerations given in [9], [10] combinations of coordinate and momenta were employed.…”

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

“…Remark 5 Simultaneous solutions of the KdV equation ( 4) and the representative of the hierarchy (23) with n = 2 and µ 2 = 0 determine simultaneous solutions (see [40]) of a pair of isomonodromic Hamiltonian systems H 7/2+1 with two degrees of freedom belonging to Kawamuko's list [41]. This list of Hamiltonian systems also consists of degenerations of the classical isomonodromic Garnier pair, but they are different from the hierarchy of its degenerations written out in [20].…”