“Quantizations” of the second Painlevé equation and the problem of the equivalence of its L-A pairs

Suleǐmanov, B. I.

doi:10.1007/s11232-008-0106-8

Cited by 29 publications

(25 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We obtained the results described before becoming aware of [11]- [13], where equations of parabolic type that are part of the Lax representation of the Painlevé I-VI equations were obtained. For comparison, we give an equivalent form of the Fuchs system [14], eliminating f from which gives the Painlevé VI equation for u.…”

Section: Systems Of Linear Equations Accompanying the Painlevé And Scmentioning

confidence: 99%

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Novikov

2009

Theor Math Phys

View full text Add to dashboard Cite

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 matrices. This generalizes Suleimanov's result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function. Systems of linear equations accompanying the Painlevé andSchlesinger equations The Schlesinger system [1] for the matriceswhere i, j = 1, m (the second group of equations can be replaced with the condition that the matrix A 1 + · · · + A m = A ∞ is constant in t 1 , . . . , t m ), was discovered as the compatibility condition for the system(2)All algebraic integrals of motion of system (1) are known. They are the traces tr A k i , k = 1, 2, . . . (equivalently, the characteristic polynomials of the matrices A i ). A closed form ω = H i dt i was found in [2],and the τ -function (log τ ) ti = H i was also determined.Here, we consider only one aspect of the problem of the complete integrability of system (1) in terms of special functions. Namely, in the case of 2×2 matrices A i , we seek a second-order linear equation for

show abstract

Section: Systems Of Linear Equations Accompanying the Painlevé And Scmentioning

confidence: 99%

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Novikov

2009

Theor Math Phys

View full text Add to dashboard Cite

show abstract

“…Another choice of order admits a symbolic writing of six linear evolution equations in form of (2) [42]. Following the terminology of paper [41], in what follows we call evolution equations (2) with constants ̸ = "quantizations" of corresponding Hamilton systems.…”

Section: Introductionmentioning

confidence: 99%

“…During the last decade, there were written quite a lot of works on relations of the equations of IDM for Painlevé type ODEs with evolution linear equations of quantum mechanics and, starting from work [36], of quantum field theory [1]- [3], [7], [8], [14]- [17], [19]- [24], [26], [27], [32], [35]- [38], [41], [43]- [45].…”

Section: Introductionmentioning

confidence: 99%

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

Pavlenko¹,

Suleǐmanov²

2017

Ufimskii Matematicheskii Zhurnal

View full text Add to dashboard Cite

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.

show abstract

“…Following the terminology of [4], we refer to equations of the type of the Schrödinger equations (2) with constants ε not depending on h as "quantizations." (Such equations with ε = 1 are used, e.g., in the description of the filtration of diffusion processes (see [5] and [6]).…”

mentioning

confidence: 99%

“…The results of papers [2] and [3] received further development in [4] and [7]- [11]. However, similar "quantizations" for higher analogues of the Painlevé equations have not been studied so far.…”

mentioning

confidence: 99%

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

Suleǐmanov

2014

Funct Anal Its Appl

Self Cite

View full text Add to dashboard Cite

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H 1 (z, t, q 1 , q 2 , p 1 , p 2 ) corresponding to the second equation P 2 1 in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P 2 1 with respect to z . This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian H 2 (z, t, q 1 , q 2 , p 1 , p 2 ) of a Hamiltonian system with respect to t compatible with P 2 1 . A similar situation occurs for the P 2 2 equation in the Painlevé II hierarchy.For all of the six Painlevé ordinary differential equations (ODEs) q zz = f (z, q, q z ), an explicit change in the solutions of the corresponding pairs of linear systems of the isomonodromic deformation method (IDM) given in Garnier's paper [1] yields solutions of the equations (see [2] and [3])(see also the beginning of the conclusion of this paper). These equations are determined by the Hamiltonians H = H(z, q, p) of Hamiltonian systems q z = H p (z, q, p) and p z = −H q (z, q, p); the Hamiltonians are quadratic in the momenta p, and eliminating p from the systems, we obtain the six Painlevé ODEs. From the Schrödinger equationswhich depend on the Planck constant h = 2π = −2πiε, the six evolution equations (1) are obtained by the formal change ε = 1. (In the cases of the Painlevé III and V equations, expressions given in [2] and [3] contain inaccuracies, which, however, are easy to correct.) Note that, for the ODEs obtained from the first and second Painlevé equations, the corresponding IDM equations can be rescaled into compatible systems of linear ODEs, which determine, for any fixed complex number ε, exact solutions of Eqs. (2) with Hamiltonians H depending on ε. Let us clarify this point. The pair of Garnier IDM equations [1] for the ODE(the a j are any constants), whose special cases are the first and second Painlevé equations, has the form W yy = P (τ, y)W, W τ = B(τ, y)W y − 1 2 B y (τ, y)W, (4) *

show abstract

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its L-A pairs

Abstract: We show that the known Flaschka-Newell L-A pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.

Cited by 29 publications

References 29 publications

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

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

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

Contact Info

Product

Resources

About

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its L-A pairs

Abstract: We show that the known Flaschka-Newell L-A pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.

Cited by 29 publications

References 29 publications

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

&#x201C;Quantizations&#x201D; of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

Contact Info

Product

Resources

About

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