Bäcklund transformations for a matrix second Painlevé equation

2010 ◽  
Vol 374 (34) ◽  
pp. 3422-3424 ◽  
Author(s):  
P.R. Gordoa ◽  
A. Pickering ◽  
Z.N. Zhu
Keyword(s):  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
Painleve Equation ◽  
Second Painlevé Equation

scholarly journals On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2095
Author(s):  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Jordan Algebra ◽  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Jordan Product ◽  
Completely Integrable ◽  
Matrix Generalization ◽  
Backlund Transformations ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Multiple Field

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation

2020 ◽  
pp. 2150029 ◽  
Author(s):  
Jie hu ◽  
Galina Filipuk ◽  
Yang Chen
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Recurrence Coefficients ◽  
Backlund Transformations ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Painleve Equation

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the [Formula: see text]-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.

scholarly journals THE BÄCKLUND TRANSFORMATIONS OF THE FIFTH PAINLEVÉ EQUATION AND THEIR APPLICATIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 221-230 ◽  
Author(s):  
V. Gromak ◽  
G. Filipuk
Keyword(s):  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Nonlinear Superposition ◽  
Backlund Transformations ◽  
Painleve Equation

The main objective of this paper is to study the properties of the Bäcklund transformations and the auto‐Bäcklund transformations, to construct the nonlinear superposition formulas for the solutions of the fifth Painlevé equation and to obtain the values of the parameters after the successive application of the Bäcklund transformations.

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