Linearizable second order Monge-Ampère equations
type: Item
instance of: scholarly article
title: Linearizable second order Monge-Ampère equations
zbMATH Open document ID: 0902.35072
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-12-02T00:00:00Z
review text: Second order nonlinear hyperbolic equations generally have solutions that develop singularities in a finite time. However, certain equations in this class are ``completely exceptional'' in that they behave like linear equations. Completely exceptional equations have been classified, and in \((1+1)\) dimensions they are Monge-Ampère equations of the form \[ \kappa_1(u_{tt}u_{xx}-u_{tx}^2) + \kappa_2 u_{tt} + \kappa_3 u_{tx} + \kappa_4 u_{xx} + \kappa_5 = 0, \] where the coefficients \(\kappa_1,\cdots,\kappa_5\) are arbitrary functions of \(x,t,u,u_t\) and \(u_x\). In \((n+1)\) dimensions there is an analogous classification. This suggests that there should be some way of transforming these equations to linear canonical forms, and indeed, some results in this direction have been obtained by \textit{A. Donato, U. Ramgulam} and \textit{C. Rogers} [Meccanica 27, 257-262 (1992; Zbl 0764.35066)]. Here the author considers the problem of linearizing the first order quasilinear systems which are equivalent to second order Monge-Ampère equations. It is shown that this can always be done in \((1+1)\) dimensions if the coefficients \(\kappa_1,\cdots,\kappa_5\) are constants, or depend only on the first derivatives of the solution. In addition, in \((2+1)\) and \((3+1)\) dimensions some conditions on the coefficents are found which guarantee that the first order system can be linearized.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35A25
zbMATH DE Number: 1130501
zbMATH Keywords: hyperbolic Monge-Ampère equations
zbMATH Keywords: linearization of first order quasilinear systems
zbMATH Keywords: completely exceptional equations
author: F. Oliveri
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5752
OpenAlex ID: W2069730059
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DOI: 10.1006/JMAA.1997.5752
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description: scientific article; zbMATH DE number 1130501