Two hierarchies of ordinary differential equations and their properties
type: Item
instance of: scholarly article
title: Two hierarchies of ordinary differential equations and their properties
zbMATH Open document ID: 0948.34069
DOI: 10.1016/S0375-9601(98)00950-5
author: Nikolai A. Kudryashov
published in: Physics Letters. A
publication date: 2000-03-08T00:00:00Z
review text: The author considers the two systems \(u_t+ \theta_1G_n(u)= 0\), and \(a_t+ \theta_2H_n(a)= 0\), \(n= 1,2,3,\dots\), with \(\theta_1= \theta_2= D^3+ 2uD+ u_x\), \(D= {\partial\over\partial x}\). The operators \(G_n\) and \(H_n\) are obtained by the recursion relations \(G_{n+2}= J_1(u)\theta_1(u) G_n\), and \(H_{n+2}= J_2(a) \theta_2(a) H_n\), when \(G_0= 1\), \(G_1= u_{xx} +{1\over 4} u^2\), \(H_0= 1\), \(H_1= a_{xx}+ 4a^2\) and the operators \(J_1= D^3+{1\over 2} D^2uD^{-1}+{1\over 2}D^{- 1} uD^2+{1\over 8}(u^2 D^{-1}+ D^{-1} u^2)\), and \(J_2= D^3+ 3(uD+ Du)+ 2(D^2 uD^{-1}+ D^{-1} uD^2)+ 8(u^2 D^{-1}+ D^{-1} u^2)\). This article is one of many by the author on this subject. These two systems are modified and when the Painlevé test is applied to these equations another system is obtained which is comparable to the system studied in an earlier paper. The author shows that this system is comparable with a desired system and has similar properties.
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 33E17
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 1412362
zbMATH Keywords: hierarchies
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: recursion relations
reviewed by: Hussain S. Nur
MaRDI profile type: Publication
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full work available at URL: s0375-9601(98)00950-5
OpenAlex ID: W2039854948
description: scientific article; zbMATH DE number 1412362