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**Double reduction of PDEs from the association of symmetries with conservation laws with applications.**
*(English)*
Zbl 1116.35004

The 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### Citations:

Zbl 0962.35009
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\textit{A. Sjöberg}, Appl. Math. Comput. 184, No. 2, 608--616 (2007; Zbl 1116.35004)

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### References:

[1] | Kara, A.; Mahomed, F., The relationship between symmetries and conservation laws, Int. J. Theor. Phys., 39, 1, 23-40 (2000) · Zbl 0962.35009 |

[2] | Stephani, H., Differential Equations: Their Solutions Using Symmetries (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.34001 |

[3] | Bluman, G.; Kumei, S., Symmetries and Differential Equations. Symmetries and Differential Equations, Graduate Texts in Mathematics, vol. 81 (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0698.35001 |

[4] | Olver, P., Applications of Lie Groups to Differential Equations. Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0588.22001 |

[5] | A. Sjöberg, Non-local Symmetries and Conservation Laws for Partial Differential Equations, Thesis, University of the Witwatersrand, Gauteng, 2002.; A. Sjöberg, Non-local Symmetries and Conservation Laws for Partial Differential Equations, Thesis, University of the Witwatersrand, Gauteng, 2002. |

[6] | Sjöberg, A.; Mahomed, F., Non-local symmetries and conservation laws for one-dimensional gas dynamics equations, Appl. Math. Comput., 150, 2, 379-397 (2004) · Zbl 1102.76059 |

[7] | Sjöberg, A.; Mahomed, F., The association of non-local symmetries with conservation laws: applications to the heat and Burger’s equations, Appl. Math. Comput., 168, 2, 1098-1108 (2005) · Zbl 1084.35075 |

[8] | Steeb, W. H.; Strampp, W., Diffusion equations and Lie and Lie-Bäcklund transformation groups, Physica, 114A, 95-99 (1982) · Zbl 0513.58045 |

[9] | A. Kara, F. Mahomed, Action of Lie-Bäcklund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997.; A. Kara, F. Mahomed, Action of Lie-Bäcklund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997. |

[10] | N. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, Chemical Rubber Company, Boka Raton, FL, 1994-1996.; N. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1-3, Chemical Rubber Company, Boka Raton, FL, 1994-1996. · Zbl 0864.35001 |

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