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Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

Published online by Cambridge University Press:  16 July 2009

G. W. Bluman  and
S. Kumei
G. W. Bluman
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, CanadaV6T 1Y4
S. Kumei
Affiliation:
Faculty of Textile Science and Technology, Shinshu University, Ueda, Nagano 386, Japan
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Abstract

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 3 , September 1990 , pp. 189 - 216
Copyright
Copyright © Cambridge University Press 1990

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