Asymptotic solutions of the second Painlevé equation in the complex plane. Inverse monodromy problem approach

2006 ◽  
pp. 261-295
Author(s):  
Athanassios Fokas ◽  
Alexander Its ◽  
Andrei Kapaev ◽  
Victor Novokshenov
Keyword(s):  
Complex Plane ◽  
Painlevé Equation ◽  
Asymptotic Solutions ◽  
Painleve Equation ◽  
Second Painlevé Equation

scholarly journals Stokes phenomena in discrete Painlevé II

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160539 ◽  
Author(s):  
N. Joshi ◽  
C. J. Lustri ◽  
S. Luu
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Complex Plane ◽  
Painlevé Equation ◽  
Asymptotic Solutions ◽  
Series Expression ◽  
Painleve Equation ◽  
Rapid Switching ◽  
Power Series Methods ◽  
Stokes Phenomena

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous 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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