Some classification results for hyperbolic equations \(F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0\)
type: Item
instance of: scholarly article
title: Some classification results for hyperbolic equations \(F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0\)
zbMATH Open document ID: 0959.35123
author: Martin Juráš
published in: Journal of Differential Equations
publication date: 2001-04-26T00:00:00Z
review text: Let \[ F(x, y, u, p, q, r, s, t)=0\tag{1} \] be a hyperbolic partial differential equation, where \(p = u_x\), \(q=u_y\), \(r=u_{xx}\), \(s=u_{xy}\), \(t=u_{yy}\). The equation (1) is said to be contact equivalent to the equation \[ F(\overline{x},\overline{y},\overline u,\overline u_{\overline x},\overline{u}_{\overline{y}},\overline{u}_{\overline{x}\overline{x}},\overline{u}_{\overline{x}\overline{y}},\overline{u}_{\overline{y}\overline{y}})=0\tag{2} \] if there is an invertible transformation \((x,y,u,u_x,u_y)\to (\overline x,\overline y,\overline u,\overline{u}_{\overline{x}}, \overline{u}_{\overline{y}})\) such that the solutions of (1) are mapped to the solutions of (2). A necessary and sufficient condition for (1) to be contact equivalent to an \(f\)-Gordon equation \[ s +f(x,y,u,p,q) = 0 \] is given in terms of the characteristic vector fields. Necessary and sufficient conditions in order that (1) is contact equivalent to the wave equation \(s = 0\) or to an \(f\)-Gordon which is linear in \(p\) and \(q\) are given in terms of the generalized Laplace invariants. The author also gives necessary and sufficient conditions in order that the equation (1) is contact equivalent to an Euler-Poisson-Darboux equation or Liouville equation or a linear equation or the Klein-Gordon equation \(s = u\). This type of results are established for a hyperbolic Monge-Ampère equation, a Fermi-Ulam-Pasta equation and a number of other equations.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 1515445
zbMATH Keywords: exact integrability
zbMATH Keywords: Darboux method
zbMATH Keywords: \(f\)-Gordon equation
zbMATH Keywords: generalized Laplace invariants
zbMATH Keywords: Euler-Poisson-Darboux equation
zbMATH Keywords: Liouville equation
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: hyperbolic Monge-Ampère equation
zbMATH Keywords: Fermi-Ulam-Pasta equation
reviewed by: Hiroki Tanabe
describes a project that uses: HELMHOLTZ
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3725
OpenAlex ID: W2083617110
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DOI: 10.1006/JDEQ.1999.3725
description: scientific article; zbMATH DE number 1515445