On special solutions to the Ermakov-Painlevé XXV equation
type: Item
instance of: scholarly article
title: On special solutions to the Ermakov-Painlevé XXV equation
zbMATH Open document ID: 1545.34114
author: A. V. Chichurin
author: Galina Filipuk
published in: Random Matrices: Theory and Applications
publication date: 2024-06-03T00:00:00Z
review text: Let \(F\) be an ordinary differential field of characteristics zero with differentiation \('=d/dz\). Denote by \(EP_{F}\) a family of differential equations of the form\N\[\N(12w+b(z))w''=15w^{,2}-h_{0}(z)w'-8w^{3}-h_{1}(z)w^{2}-h_{2}(z)w-h_{3}(z),\N\]\Nwhere \(b,h_{0},\ldots,h_{3}\in F.\) Similarly, introduce the notation \(\mathcal{L_{\textrm{F}}}\) and \(E_{F}\) for the families of equations \N\[\Ny'''+p(z)y''+q(z)y'+r(z)y=0 \text{ and } y''+\Phi(z)y=ay^{-3},\N\] \Nrespectively. The paper discusses the possibility of reduction of equations from \(EP_{F}\) to equations from \(\mathcal{L_{\textrm{F}}}\) and \(E_{F}\). In other words, there are selected such non-trivial subsets of equations in the family \(EP_{F}\), which can be differently-rationally maped to the corresponding subsets in \(\mathcal{L_{\textrm{F}}}\) or \(E_{F}\). For example, as it is shown in the paper, a one-parameter subset from \(EP_{F}\)\N\[\N(3(k^{''}+2w^{''})+2(2k+w)(k+2w))(k+2w)-\dfrac{15}{4}(k^{'}+2w^{'})^{2}=0\:(k\in F),\N\]\Nis maped on the \(E_{F}\) thus\N\[\Nu^{''}=\dfrac{1}{4}ku+\dfrac{\lambda}{72}u^{-3}(k\in F,\lambda=\mathrm{const}).\N\]\NHowever, the selection of the corresponding subset for the case \(\mathcal{L_{\textrm{F}}}\) is a non-trivial task, despite the presence of a differential-regular map \(\mathcal{L_{\textrm{F}}}\) in \(EP_{F}\). The authors describe their ideas to solve this problem.
reviewed by: Mykola Grygorenko
Mathematics Subject Classification ID: 34M04
Mathematics Subject Classification ID: 34M03
Mathematics Subject Classification ID: 34M25
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 7859064
zbMATH Keywords: Riccati equation
zbMATH Keywords: generalized Riccati equation
zbMATH Keywords: Painlevé XXV equation
zbMATH Keywords: Ermakov equation
zbMATH Keywords: special functions
zbMATH Keywords: linear differential equations
MaRDI profile type: Publication
cites work: NIST digital library of mathematical functions
cites work: Q3331506
cites work: Q3665386
cites work: On Dirichlet two-point boundary value problems for the Ermakov-Painlevé IV equation
cites work: On a Neumann boundary value problem for Ermakov–Painlevé III
cites work: On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem
cites work: Ermakov-Pinney and Emden-Fowler equations: new solutions from novel Bäcklund transformations
cites work: A short note on the Painlevé XXV-Ermakov equation
cites work: Differential identities for the structure function of some random matrix ensembles
cites work: Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE
cites work: Mixed superposition rules and the Riccati hierarchy
cites work: A Lie systems approach to the Riccati hierarchy and partial differential equations
cites work: Q3231543
cites work: The Ermakov equation: A commentary
cites work: On third-order linear equations
cites work: Q4948208
cites work: Q3564676
cites work: Q3151170
cites work: Linear differential equations for the resolvents of the classical matrix ensembles
cites work: A novel Ermakov-Painlevé II system: \(N+1\)-dimensional coupled NLS and elastodynamic reductions
cites work: Hybrid Ermakov-Painlevé IV Systems
cites work: On Ermakov-Painlevé II systems. Integrable reduction
cites work: On Hybrid Ermakov-Painlevé Systems. Integrable Reduction
cites work: On integrable Ermakov-Painlevé IV systems
cites work: Ermakov-Painlevé II symmetry reduction of a Korteweg capillarity system
cites work: Ermakov-Painlevé II Reduction in Cold Plasma Physics. Application of a Bäcklund Transformation
cites work: Differential equations for singular values of products of Ginibre random matrices
cites work: Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians
DOI: 10.1142/S2010326324500011
description: scientific article; zbMATH DE number 7859064