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Contact integrable extensions of symmetry pseudo-groups and coverings of \((2+1)\) dispersionless integrable equations. (English) Zbl 1186.58023

Many useful methods for PDE’s can be successfully described in the framework of the theory of coverings [I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math. 2, 79–96 (1984; Zbl 0547.58043)]. The aim of this paper is to present a systematic approach to the method of determining coverings of PDE’s from invariant linear combinations of Maurer-Cartan forms of their contact symmetry pseudo-groups, even for PDE’s with more than two independent variables, when the symmetry pseudo-group is infinite dimensional.
The approach is based on contact integrable extensions, so the authors propose a generalization of the definition of an integrable extension of an exterior differential system [R. L. Bryant and P. A. Griffiths, Duke Math. J. 78, No. 3, 531–676 (1995; Zbl 0853.58005)] to the case of more than two independent variables.
“The theory of integrable extensions is then applied to the symmetry pseudo-groups of the \(r\)-th mdKP equation, the \(r\)-th dDym equation, and the deformed Boyer-Finley equation. This gives another look at deriving known coverings and leads to new coverings for these equations.”

MSC:

58J70 Invariance and symmetry properties for PDEs on manifolds
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
58H05 Pseudogroups and differentiable groupoids
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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