Generalized Laplace invariants and the method of Darboux
type: Item
instance of: scholarly article
title: Generalized Laplace invariants and the method of Darboux
zbMATH Open document ID: 0885.35075
DOI: 10.1215/S0012-7094-97-08916-X
author: Martin Juráš
author: I. M. Anderson
published in: Duke Mathematical Journal
publication date: 1998-04-23T00:00:00Z
review text: In rough but suggestive terms, the authors transparently demonstrate the important idea that certain constructions as yet well-known only for the linear differential equations can be carried over to the nonlinear case if appropriate differential forms undertake the role of functions. In this sense, the familiar Laplace method which permits to resolve some linear differential equations \(u_{xy}+ a(x,y)u_x+ b(x,y)u_y+ c(x,y)u= 0\) can be adapted in such a manner to give the classical Darboux method for the hyperbolic equations \(F(x,y,u, u_x,u_y, u_{xx}, u_{xy}, u_{yy})= 0\). The rather involved calculations are based upon the so-called adapted Laplace coframe (the counterpart of the Laplace series of transformations in the linear subcase) which is thoroughly analyzed and illustrated by the examples of equations \(u_{xx} u_{yy}= u_y\), \(u_{xy}+ uu_{xx}= f(x,u_x)\), \(u_{xx}= f(u_{yy})\).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 58J72
zbMATH DE Number: 1061778
zbMATH Keywords: differential forms
zbMATH Keywords: adapted Laplace coframe
reviewed by: Jan Chrastina
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1061778