On the integrability of non-polynomial scalar evolution equations
type: Item
instance of: scholarly article
title: On the integrability of non-polynomial scalar evolution equations
zbMATH Open document ID: 0959.35147
author: Jan A. Sanders
author: Jing Ping Wang
published in: Journal of Differential Equations
publication date: 2001-04-26T00:00:00Z
review text: The (symmetry) integrability of a nonlinear evolution partial differential equation (in 1+1 dimensions: time plus space) is considered tantamount to possession of an infinite number of symmetries. The equations under consideration are classified by a grading prescription (\(\lambda\)-homogeneity). In a previous paper the authors had ``produced the complete list of \(\lambda\)-homogeneous (symmetry) integrable equations with positive \(\lambda\)''. The main result of this paper is to ``give the complete list of 0-homogeneous integrable systems''. It turns out that such integrable equations are equivalent up to homogeneous transformations to the equations contained in hierarchies of: Heat Equation/potential Burgers equation \(u_t=u_2\); Potential modified Korteweg-de Vries equation, \(u_t=u_3+u^3_1\); Potential Kupershmidt equation, \(u_t=u_5+5u_2u_3-5u_1u^2_2+u^5_1\). (The reader will note that these equations are all polynomial; but nonpolynomial equations are also considered in the paper, justifying its title). The authors also ``draw some conclusions from these results for the case \(\lambda=-1\) which, although theoretically incomplete, seem to cover the known integrable systems for this case''.
reviewed by: Francesco Calogero
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 1517169
zbMATH Keywords: integrability
zbMATH Keywords: symmetries
zbMATH Keywords: classification
describes a project that uses: Maple
describes a project that uses: FORM
MaRDI profile type: Publication
OpenAlex ID: W2080104147
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DOI: 10.1006/JDEQ.2000.3782
description: scientific article; zbMATH DE number 1517169