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A hierarchy of submodels of differential equations

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Abstract

We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.

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Authors and Affiliations

  1. Institute of Mechanics of the Ufa Scientific Center of the Russian Academy of Sciences, Ufa State Aviation Technical University, Ufa, Russia

    S. V. Khabirov

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  1. S. V. Khabirov
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Correspondence to S. V. Khabirov.

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Original Russian Text Copyright © 2013 Khabirov S.V.

The author was supported by the Russian Foundation for Basic Research (Grants 11-01-00026-a, 11-01-00147-a, and 12-01-00648), the Federal Agency for Science and Innovation of the Russian Federation (Grant NSh-6706.2012.1), and the Government of the Russian Federation (Grant 11.G34.31.0042, Resolution No. 220).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 6, pp. 1396–1406, November–December, 2013.

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Khabirov, S.V. A hierarchy of submodels of differential equations. Sib Math J 54, 1110–1119 (2013). https://doi.org/10.1134/S0037446613060189

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  • DOI: https://doi.org/10.1134/S0037446613060189

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