Special polynomials and rational solutions of the hierarchy of the second Painlevé equation

2007 ◽  
Vol 153 (1) ◽  
pp. 1398-1406 ◽  
Author(s):  
M. V. Demina ◽  
N. A. Kudryashov
Keyword(s):  
Painlevé Equation ◽  
Rational Solutions ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

scholarly journals Remarks on the Yablonskii-Vorob’ev polynomials

2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda
Keyword(s):  
Algebraic Approach ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Special polynomials associated with rational solutions of the defocusing nonlinear Schrödinger equation and the fourth Painlevé equation

2006 ◽  
Vol 17 (3) ◽  
pp. 293-322 ◽  
Author(s):  
PETER A. CLARKSON
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Painlevé Equation ◽  
Rational Solutions ◽  
Oscillatory Solutions ◽  
Nonlinear Schrödinger ◽  
Special Polynomials ◽  
Painleve Equation

Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions

2012 ◽  
Vol 468 (2146) ◽  
pp. 3247-3264 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Linear Problem ◽  
Infinite Sequence ◽  
Painlevé Equation ◽  
Rational Solutions ◽  
Determinantal Representation ◽  
Deformation Problem ◽  
Hypergeometric Type ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Hypergeometric Solutions

This is the second part of our study of the solutions of a q -discrete second Painlevé equation ( q -P II ) of type ( A 2 + A 1 ) (1) via its iso-monodromy deformation problem. In part I, we showed how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions. In this paper, we study the case giving rise to an infinite sequence of q -hypergeometric-type solutions. We find a new determinantal representation of all such solutions and solve the iso-monodromy deformation problem in closed form.

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.

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