Darboux integrable nonlinear hyperbolic equations
type: Item
instance of: scholarly article
title: Darboux integrable nonlinear hyperbolic equations
zbMATH Open document ID: 0880.35077
author: O. Diekmann
published in: Doklady Mathematics
publication date: 1998-01-28T00:00:00Z
review text: We study equations with two independent variables of the form \[ u_{xy}=F(x,y,u,u_x,u_y).\tag{1} \] It is well-known that there exist two types of integrable equations (1). The first type includes the wave equation, the Liouville equation \(u_{xy}=\exp(u)\), and numerous analogs of the latter. The best-known equation of the second type is the sine-Gordon equation \(u_{xy}=\sin(u)\). Equations of the first type are called Darboux integrable equations, because Darboux appears to be the first to have defined them rigorously and to have suggested a geometric method for their integration. The problem of a complete classification of such equations remains unsolved. We establish a connection between Darboux integrable equations (1) and differentiable substitutions of the type of the Miura transform for evolutionary equations; we suggest an effective criterion of the integrability according to Darboux in terms of the Laplace invariants for the linearization operator and indicate an explicit formula for higher symmetries of Darboux integrable equations (1).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 1050605
zbMATH Keywords: Liouville equation
zbMATH Keywords: Miura transform
zbMATH Keywords: Laplace invariants
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1050605