Invariant solutions of a system of Euler equations that satisfy the Rankine–Hugoniot conditions

1Yuryk, II
1National University of Food Technologies, Kyiv
Dopov. Nac. akad. nauk Ukr. 2018, 7:10-19
https://doi.org/10.15407/dopovidi2018.07.010
Section: Mathematics
Language: English
Abstract: 

We consider equations of hydrodynamics with certain additional constraints. Group-theoretical methods are applied to find invariant solutions of a system of Euler equations that satisfy the Rankine–Hugoniot conditions.

Keywords: Euler equations, group-theoretical methods, invariant solutions
References: 
  1. Olver, P. (1986). Applications of Lie groups to differential equations. New York: Springer. doi: https://doi.org/10.1007/978-1-4684-0274-2
  2. Ovsjannikov, L. V. (1978). Group analysis of differential equations. Moscow: Nauka (in Russian).
  3. Ovsjannikov, L. V. (1981). Lectures on the fundamentals of gas dynamics. Moscow: Nauka (in Russian).
  4. Fushchych, W. I. (1981). Symmetry in problems of mathematical physics. In Algebraic-theoretical studies in mathematical physics (pp. 6-28). Kiev (in Russian).
  5. Barannyk, A. F. & Yuryk, I. I. (1998). On a new method for constructing exact solutions of the nonlinear differential equations of mathematical physics. J. Phys. A: Math. Gen., 31, No. 21, pp. 4899-4907. doi: https://doi.org/10.1088/0305-4470/31/21/008
  6. Barannyk, A. F., Barannyk, T. F. & Yuryk, I. I. (2011). Separation of variables for nonlinear equations of hyperbolic and Korteweg–de Vries type. Rep. Math. Phys., 68, No. 1, pp. 97-105. doi: https://doi.org/10.1016/S0034-4877(11)60029-3
  7. Barannyk, A. F., Barannyk, T. A. & Yuryk, I. I. (2013). On hidden symmetries and solutions of the nonlinear d'Alembert equation. Commun. Nonlinear Sci. Numer. Simul., 18, No. 7, pp. 1589-1599. doi: https://doi.org/10.1016/j.cnsns.2012.11.013
  8. Bihlo, A. & Popovych, R. O. (2009). Lie symmetries and exact solutions of barotropic vorticity equation. J. Math. Phys., 50, 123102, 12 pp. doi: https://doi.org/10.1063/1.3269919
  9. Fushchych, W. I. & Serova, M. M. (1983). On the maximal invariance group and general solution of the onedimensional gas dynamics equations. Dokl. AN SSSR, 268, No. 5, pp.1102-1104 (in Russian).
  10. Korobeynikov, V. P., Mel'nikova, N. S. & Ryazanov, E. V. (1961). The theory of point explosion. Moscow: Fizmatgiz (in Russian).
  11. Yuryk, I. I. (2015). Application of group-theoretical methods to solving the point explosion problem in incompressible liquid. Commun. Nonlinear Sci. Numer. Simul., 22, pp. 1017-1027, doi: https://doi.org/10.1016/j.cnsns.2014.09.022