Criteria for the existence of Bäcklund transformations
A pedagogical guide to the meaning and purpose of Bäcklund transformations is given. Jet bundles of C∞ maps of manifolds are used to describe partial differential equations (PDE's) and various kinds of transformations on them. The purposes are twofold. First, to explain to the non-expert, the basic ideas of this approach via intuitive concepts. Second, to describe recently-developed criteria which can be used to determine the nonexistence of a Bicklund transformation for any particular equation or system of equations. Previous work on Bäcklund transformations is extended to the case of non-quasi-linear PDE's in an arbitrary number of independent variables. The general form (modulo contact transformations) is given that a PDE must have in order that a Bäcklund transform of it may exist. For more than two independent variables, the concept of multiple copies of the original equation is introduced and an explanation for the usefulness of this concept is sketched. As an indication of the utility of such a negative concept, it is also explained how the author came to be interested in such transforms -certain equations which occur in the study of self-dual Einstein spaces- and the conclusions which the above considerations lead to: These equations do not admit Bäcklund transformations.
KeywordsLaplace Equation Integrability Condition Nonlinear Evolution Equation Base Manifold Contact Transformation
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