Double reduction of PDEs from the association of symmetries with conservation laws with applications.

*(English)*Zbl 1116.35004The association of conservation laws with Noether symmetries extended to Lie-Bäcklund and nonlocal symmetries has opened the possibilities to the extension of the theory on double reductions to partial differential equations that do not have a Lagrangian and therefore to not posses Noether symmetries. at the usage of the results [A. Kara, F. Mahomed, Int. J. Theor. Phys. 39, No. 1, 23–40 (2000; Zbl 0962.35009)] the author develops the theory to effect a double reduction of PDEs with two independent variables, which is possible when the PDEs admit a symmetry associated with a conservation law. This theory is illustrated by applications to the linear heat equation, the sine-Gordon and BBM equations and a system of PDEs from one dimensional gas dynamics.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35L65 | Hyperbolic conservation laws |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

##### Keywords:

partial differential equations; Lie point symmetries; conservation laws; double reduction; heat equation; BBM equation; sine-Gordon equation; one dimensional gas dynamics
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\textit{A. Sjöberg}, Appl. Math. Comput. 184, No. 2, 608--616 (2007; Zbl 1116.35004)

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##### References:

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