Geodesic
Simple solutions of rank 2 invariant submodel for monatomic gas
Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 3, pp. 353-373.

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We consider the invariant submodel of rank 2 on two-dimensional subalgebra containing the projective operator and the dilatation operator with respect to the thermodynamic parameters of a monatomic gas. The submodel is written in the canonical form, therefore, the representation of the solution has a nontrivial form. Invariant solutions, irrotational motions, and a series of simple solutions with differential relations are obtained. The gas motions are without collapse on the whole space. Gas is compressed at first, and then it is dispersed to the vacuum. Soliton-like movements of gas bunch have also been found.
Keywords: monatomic gas, invariant submodel, differential relations, solitary density waves. 
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S. V. Khabirov; R. F. Shayakhmetova. Simple solutions of rank 2 invariant submodel for monatomic gas. Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 3, pp. 353-373. http://geodesic.mathdoc.fr/item/CHFMJ_2018_3_3_a8/

[1] Ovsyannikov L.V., “The "podmodeli"\: program. Gas dynamics”, Journal of Applied Mathematics and Mechanics, 58:4 (1994), 601–627 | DOI | MR | Zbl

[2] Chirkunov Yu.A., Khabirov S.V., Elements of differential equations symmetry analysis for continuum mechanics, Novosibirsk State Technical University, Novosibirsk, 2012, 659 pp. (In Russ.)

[3] Cherevko A.A., The optimal system of subalgebras for the Lie algebra of operators admissible by the system of equations of gas dynamics with the equation of state $p=f(S)\rho^{5/3} $, no. 4, Siberian Branch of RAS, Institute of Hydrodynamics, Novosibirsk, 1996, 39 pp.

[4] Shayakhmetova R.F., “Inserted invariant submodels for motion of monatomic gas”, Siberian Electronic Mathematical Reports, 11 (2014), 605–625 (In Russ.) | MR | Zbl

[5] Khabirov S.V., “A hierarchy of submodels of differetial equations”, Siberian Mathematical Journal, 54:6 (2013), 1110–1119 | DOI | MR | Zbl

[6] Khabirov S.V., “Reduction of an invariant submodel of gas dynamics to canonical form”, Mathematical Notes, 66:3–4 (1999), 355–359 | DOI | DOI | MR | Zbl

[7] Shayakhmetova R.F., “Invariant submodels of rank 3 and rank 2 monatomic gas with the projective operator”, Proceedings of the Mavlyutov Institute of Mechanics of Ufa Scientific Center of RAS, 11, no. 1, 2016, 127–135 (In Russ.)

[8] Shayakhmetova R.F., “The compression of gas followed by expansion”, Proceedings of the Mavlyutov Institute of Mechanics of Ufa Scientific Center of RAS, 12, no. 2, 2017, 195–198 (In Russ.)