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Symmetry-based algorithms to relate partial differential equations. I: Local symmetries. (English) Zbl 0718.35003
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. [For part II see ibid. 217-223 (1990; Zbl 0718.35004).]
Reviewer: G.W.Bluman

MSC:
35A30 Geometric theory, characteristics, transformations in context of PDEs
35G20 Nonlinear higher-order PDEs
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
17B99 Lie algebras and Lie superalgebras
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References:
[1] DOI: 10.1137/0143084 · Zbl 0544.35007
[2] DOI: 10.1016/0022-247X(90)90431-E · Zbl 0696.35005
[3] DOI: 10.1007/BF01443337 · JFM 07.0233.02
[4] Varley, Stud. Appl. Math. 72 pp 241– (1985) · Zbl 0575.35006
[5] Sukharev, Dokl. Akad. Nauk USSR 175 pp 781– (1967)
[6] Ovsiannikov, Group Analysis of Differential Equations (1982)
[7] Olver, Applications of Lie Groups to Differential Equations (1986) · Zbl 0588.22001
[8] DOI: 10.1063/1.1664700 · Zbl 0283.35018
[9] DOI: 10.1007/BF01443183 · JFM 07.0233.03
[10] Lie, Theorie der Transformationsgruppen II (1890)
[11] DOI: 10.1137/0142079 · Zbl 0506.35003
[12] DOI: 10.1063/1.527974 · Zbl 0669.58037
[13] Forsyth, Theory of Dfferential Equations v pp 318– (1906)
[14] Bluman, Symmetries and Differential Equations. Appl. Math. Sci. (1989) · Zbl 0698.35001
[15] Bluman, Similarity Methods for Differential Equations. Appl. Math. Sci. (1974) · Zbl 0292.35001
[16] DOI: 10.1137/0139021 · Zbl 0448.60056
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