scholarly journals Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation

2007 ◽  
Vol 32 (1) ◽  
pp. 124-144 ◽  
Author(s):  
Maria V. Demina ◽  
Nikolai A. Kudryashov
Keyword(s):  
Fourth Order ◽  
Painlevé Equation ◽  
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.

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.

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