Nonlinear differential equations associated with the first Painlevé hierarchy
type: Item
instance of: scholarly article
title: Nonlinear differential equations associated with the first Painlevé hierarchy
zbMATH Open document ID: 1414.34071
author: Nikolai A. Kudryashov
published in: Applied Mathematics Letters
publication date: 2019-02-20T00:00:00Z
review text: An integrable hierarchy associated with the first Painleve equation was introduced in the author's work [Phys. Lett., A 224, No. 6, 353--360 (1996; Zbl 0962.35504)]. It takes the form \[ L_n[w] = z, \,n = 1, 2, . . ., \] where \(z\) is the independent variable, \(w(z)\) is the dependent variable and the operator \(L_n[w]\) is determined by formula \[ \frac{d L_n[w]}{dz} =\Bigg(\frac{d^3}{dz^3}-4w \frac{d}{d^z}-2 \frac{dw}{dz}\Bigg) L_{n-1}[w],\,\, L_0[w]=-1/2. \] The case \(n=2\) corresponds to the original first Painleve equation. This hierarchy can be obtained as a compatibility condition for the following linear system \[ \Psi_{zz}=U(z,\lambda)\Psi,\,\,\, \omega(\lambda)\Psi_{\lambda}=2A(z,\lambda)\Psi_{z}-A_z(z,\lambda)\Psi. \] In the paper under review, the Cauchy problem for the equations of the hierarchy are solved by means of the inverse isomonodromic transform for the associated linear system.
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 7026125
zbMATH Keywords: first Painlevé hierarchy
zbMATH Keywords: Painlevé transcendent
zbMATH Keywords: transformation
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
full work available at URL: j.aml.2018.11.013
OpenAlex ID: W2901145493
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DOI: 10.1016/J.AML.2018.11.013
description: scientific article; zbMATH DE number 7026125