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Construction of Conservation Laws Using Symmetries

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Similarity and Symmetry Methods

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 73))

Abstract

The concept of nonlinear self-adjointness of differential equations, introduced by the author in 2010, is discussed in detail. All linear equations and systems are nonlinearly self-adjoint. Moreover, the class of nonlinearly self-adjoint equations includes all nonlinear equations and systems having at least one local conservation law. It follows, in particular, that the integrable systems possessing infinite set of Lie-Bäcklund symmetries (higher-order tangent transformations) are nonlinearly self-adjoint. An explicit formula for conserved vectors associated with symmetries is provided for all nonlinearly self-adjoint differential equations and systems. The number of equations contained in the systems under consideration can be different from the number of dependent variables. A utilization of conservation laws for constructing exact solutions is discussed and illustrated by computing non-invariant solutions of the Chaplygin equations in gas dynamics.

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Notes

  1. 1.

    See [2]. An approach in terms of variational principles is developed in [5].

  2. 2.

    He mentions in his paper that the work has been done in 1950 and published in Report # 336 of the Institute of Chemical Physics of the USSR Academy of Sciences.

  3. 3.

    Weymann uses Dreicer’s kinetic equation [20] for a photon gas interacting with a plasma which is slightly different from the equation used by Kompaneets.

  4. 4.

    The operator identity (179) was called in [31] the Noether identity and used for simplifying the proof of Noether’s theorem. A simple proof of the identity (179) can be found in [32] where Eq. (179) is written as Eq. (19).

  5. 5.

    One can verify that if one uses the canonical form of the operator \(X,\) then the operator identity (179) becomes identical with Eqs. (3), (6) in Noether’s paper [29] except for the notation. Noether comments that in the case of the first-order Lagrangians her Eq. (3) is identical with the central equation of Lagrange (Eqs. (4) and (5) in [29]).

  6. 6.

    This statement is applicable to nonlinear ODEs as well, see [35].

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Acknowledgments

I am cordially grateful to my wife Raisa for reading the manuscript carefully. Without her help and questions the work would be less palatable and contain essentially more misprints. My thanks are due to Sergey Svirshchevskii for a discussion of the material presented in Sect. 2.1. I acknowledge a financial support of the Government of Russian Federation through Resolution # 220, Agreement # 11.G34.31.0042.

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  1. Laboratory “Group Analysis of Mathematical Models in Natural and Engineering Sciences”, Ufa State Aviation Technical University, Ufa, 450 000, Russia

    Nail H. Ibragimov

  2. Research Centre ALGA: Advances in Lie Group Analysis, Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79, Karlskrona, Sweden

    Nail H. Ibragimov

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Correspondence to Nail H. Ibragimov .

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  1. LEMTA—Université de Lorraine, Vandoeuvre, France

    Jean-François Ganghoffer

  2. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivaïlo Mladenov

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Ibragimov, N.H. (2014). Construction of Conservation Laws Using Symmetries. In: Ganghoffer, JF., Mladenov, I. (eds) Similarity and Symmetry Methods. Lecture Notes in Applied and Computational Mechanics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-08296-7_2

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