Properties of solutions of two second-order differential equations with the Painlevé property
type: Item
instance of: scholarly article
title: Properties of solutions of two second-order differential equations with the Painlevé property
zbMATH Open document ID: 1476.37079
DOI: 10.1134/S0040577921030041
published in: Theoretical and Mathematical Physics
publication date: 2021-06-29T00:00:00Z
review text: The author considers a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with respect to another. He obtains two Bäcklund transformations (direct and inverse) for the solutions of the Painlevé XXXIV equation. Based on this, a nonlinear functional relation for the solutions of the Painlevé XXXIV equation with different values of its parameter is obtained. Further, he constructs a second-degree and second-order nonlinear differential equation with an arbitrary analytic function \(F(t)\) and an arbitrary parameter $\gamma$ that is a Painlevé-type equation, which for $\gamma=1$ is the canonical equation XXVII in the Ince list in the case $m=2$. Precisely, he obtains a Painlevé-type equation that reduces to the above mentioned equation for $F(t)=-t$ and $\gamma=0$. He finally shows that the direct and inverse Bäcklund transformations coincide with the pair of Bäcklund transformations for the Painlevé XXXIV equation.
reviewed by: Mengkun Zhu
Mathematics Subject Classification ID: 37J65
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34M25
zbMATH DE Number: 7364270
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Painlevé property
zbMATH Keywords: direct Bäcklund transformation
zbMATH Keywords: inverse Bäcklund transformation
author: V. V. Tsegel'nik
MaRDI profile type: Publication
Wikidata QID: Q115248080
cites work: The isomonodromic deformation method in the theory of Painlevé equations
cites work: Q5394330
cites work: Random unitary matrices, permutations and Painlevé
cites work: Polynomial Hamiltonians associated with Painleve equations. II: Differential equations satisfied by polynomial Hamiltonians
cites work: A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI
cites work: Q4881160
cites work: ``Quantizations'' of the second Painlevé equation and the problem of the equivalence of its \(L-A\) pairs
cites work: ON A NOVEL CLASS OF INTEGRABLE ODEs RELATED TO THE PAINLEVÉ EQUATIONS
cites work: Q5843703
cites work: Q5516286
cites work: Hamiltonians associated with the sixth Painlevé equation
cites work: Hamiltonians associated with the third and fifth Painlevé equations
cites work: ``Quantizations'' of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom
full work available at URL: s0040577921030041
OpenAlex ID: W3176453974
description: scientific article; zbMATH DE number 7364270