Abstract
A symmetry of a PDE is a transformation (mapping) of its solution manifold into itself, i.e., it is a transformation that maps any solution of the PDE into another solution of the same PDE. Invariant solutions (similarity solutions) are solutions that map into themselves. If a symmetry of a given PDE is a point symmetry, then invariant solutions arise constructively from a reduced differential equation with fewer independent variables [Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Bluman & Anco (2002); Cantwell (2002)].
In this chapter, we consider the problem of determining whether there exists a mapping of a given PDE into a target PDE of interest and to construct such a mapping when it exists. A target PDE is either a specific PDE or a member of a class of PDEs. The target PDE is locally equivalent to the given PDE if the mapping is invertible. The invertible mapping is not necessarily unique if a target PDE is a member of a class of PDEs. It is shown that the situation for showing existence and then finding such a mapping is especially fruitful when the target PDE (or target class of PDEs) is completely characterized by a class of contact symmetries (which only exist as point symmetries in the case of a system of PDEs).
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© 2010 Springer-Verlag New York
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Bluman, G.W., Cheviakov, A.F., Anco, S.C. (2010). Construction of Mappings Relating Differential Equations. In: Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences, vol 168. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68028-6_2
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DOI: https://doi.org/10.1007/978-0-387-68028-6_2
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Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98612-8
Online ISBN: 978-0-387-68028-6
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