Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation

2021 ◽  
Vol 120 ◽  
pp. 107248
Author(s):  
Anton Dzhamay ◽  
Galina Filipuk ◽  
Adam Ligȩza ◽  
Alexander Stokes
Keyword(s):  
Differential System ◽  
Hamiltonian Structure ◽  
Painlevé Equation ◽  
Painleve 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 New contiguity relation of the sixth Painlevé equation from a truncation

2002 ◽  
Vol 161 (3-4) ◽  
pp. 129-141 ◽  
Author(s):  
Robert Conte ◽  
Micheline Musette
Keyword(s):  
Painlevé Equation ◽  
Painleve Equation
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