Method of Conservation Laws for Constructing Solutions to Systems of PDEs

N.H. Ibragimov

Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Method of Conservation Laws for Constructing Solutions to Systems of PDEs

Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 353--365 | DOI:10.5890/DNC.2012.09.002

N.H. Ibragimov

Laboratory “Group analysis of mathematical models in natural and engineering sciences”, Ufa State Aviation Technical University, 450 000 Ufa, Russia

Center ALGA, Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden

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Abstract

In the present paper, a new method is proposed for constructing exact solutions for systems of nonlinear partial differential equations. It is called the method of conservation laws. Application of the method to the Chaplygin gas allowed to construct new solutions containing several arbitrary parameters. It is shown that these solutions cannot be obtained, in general, as group invariant solutions.

Acknowledgments

I acknowledge the financial support of the Government of Russian Federation through Resolution No. 220, Agreement No. 11.G34.31.0042.

References

[1] Laplace, P.S. (1798), Traité de Méchanique Céleste, t. 1, Paris. Reprinted in Laplace, P.S. (1878) Ouvres complétes, t. I, Gauthier-Villars, Paris. English transl., New York, 1966.
[2] Ibragimov, N.H. (1983), Transformation Groups Applied to Mathematical Physics, Nauka, Moscow. English transl. Reidel, Dordrecht, 1985.
[3] Ibragimov, N.H. (2010-2011), Nonlinear self-adjointness in constructing conservation laws, Arch. ALGA, 7/8, 1-99. See also arXiv:1109.1728v1[math-ph] (2011) 1-104.

[4]	Akhatov, I.S., Gazizov, R.K., and Ibragimov, N.H. (1989), "Nonlocal symmetries: Heuristic approach," Itogi Nauki i Tekhniki. Sovremennie problemy matematiki. Noveishye dostizhenia, 34, 3-84. English transl., 1991, Journal of Soviet Mathematics, 55(1), 1401-1450.