Equivalence of the Darboux and Gardner methods for integrating hyperbolic equations in the plane
type: Item
instance of: scholarly article
title: Equivalence of the Darboux and Gardner methods for integrating hyperbolic equations in the plane
zbMATH Open document ID: 0823.35121
DOI: 10.1215/S0012-7094-95-07707-2
author: David Hartley
author: R. W. Tucker
author: Philip A. Tuckey
published in: Duke Mathematical Journal
publication date: 1995-11-01T00:00:00Z
review text: Recall that the classical Darboux method can be applied if there exists a pair of functions \(f\), \(g\) which are constant on one of the Monge characteristics; then the intermediate integral \(\varphi (f)= \psi(g)\) together with the original differential equation have a Cauchy characteristic vector field (for arbitrary fixed \(\varphi\), \(\psi\)) and so the problem is reduced to ordinary differential equations. On the other hand, the recent Gardner method deals with a Pfaffian system \({\mathcal S}\) and directly searches for a vector field \(X\) which is Cauchy characteristic for an appropriately extended Pfaffian system \({\mathcal S}' \supset {\mathcal S}\); such \(X\) generates solutions of \({\mathcal S}'\), hence of \({\mathcal S}\). The authors prove that the Gardner method applied to a prolongation of a hyperbolic equation \(F(x,y, u,u_ x, u_ y, u_{xx}, u_{xy}, u_{yy})=0\) is equivalent to the Darboux method for its next prolongation. Lucid exposition is concluded with examples \((x+y) u_{xy}+ n(u_ x+ u_ y)=0\), \(y_{yy}= k(u_ x) u_{xx}\).
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 755407
zbMATH Keywords: Darboux method
zbMATH Keywords: Cauchy characteristic vector field
zbMATH Keywords: Pfaffian system
zbMATH Keywords: Gardner method
reviewed by: Jan Chrastina
MaRDI profile type: Publication
cites work: The cauchy problem for pfaffian systems
cites work: A differential geometric generalization of characteristics
cites work: Characteristics and the geometry of hyperbolic equations in the plane
cites work: Q3997815
cites work: An obstruction to the integrability of a class of nonlinear wave equations by 1-stable Cartan characteristics
full work available at URL: s0012-7094-95-07707-2
OpenAlex ID: W1961156684
description: scientific article; zbMATH DE number 755407