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Symmetry, Compatibility and Exact Solutions of PDEs

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Abstract

We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions. Finally we discuss models of equations with large symmetry algebras, which eventually lead to integration in closed form.

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Notes

  1. In loc. cited only the classical Monge case p=1 (1 equation on 2 unknowns) was treated, but the general case is similar.

  2. One also need to change y↦−y to match the sign.

  3. Outside the submanifold in \(\mathcal{E}_{\epsilon}\) given by Cartan equation (8)!

  4. This means the corresponding Nijenhuis tensor N J is non-degenerate.

  5. It has Lie class ω=1, i.e. the solutions depend upon 1 function of 1 argument.

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Acknowledgements

I thank V. Lychagin and A. Prasolov for helpful discussions. Some calculations in Sect. 4 were performed with Maple package DifferentialGeometry by I. Anderson. I am grateful for organizers of the conference SPT-2011 for hospitality.

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  1. Institute of Mathematics and Statistics, University of Tromsø, Tromsø, 90-37, Norway

    Boris Kruglikov

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Kruglikov, B. Symmetry, Compatibility and Exact Solutions of PDEs. Acta Appl Math 120, 219–236 (2012). https://doi.org/10.1007/s10440-012-9708-0

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