SIGMA 3 (2007), 116, 11 pages      arXiv:0709.1053      https://doi.org/10.3842/SIGMA.2007.116
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Exact Solutions of the Equations of Relativistic Hydrodynamics Representing Potential Flows

Maxim S. Borshch and Valery I. Zhdanov
National Taras Shevchenko University of Kyiv, Ukraine

Received September 10, 2007, in final form November 28, 2007; Published online December 07, 2007

Abstract
We use a connection between relativistic hydrodynamics and scalar field theory to generate exact analytic solutions describing non-stationary inhomogeneous flows of the perfect fluid with one-parametric equation of state (EOS) p = p(ε). For linear EOS p = κε we obtain self-similar solutions in the case of plane, cylindrical and spherical symmetries. In the case of extremely stiff EOS (κ = 1) we obtain ''monopole + dipole'' and ''monopole + quadrupole'' axially symmetric solutions. We also found some nonlinear EOSs that admit analytic solutions.

Key words: relativistic hydrodynamics; exact solutions.

pdf (249 kb)   ps (166 kb)   tex (29 kb)

References

1. Landau L.D., On the multiple production of particles in fast particle collisions, Izv. AN SSSR Ser. Fiz. 17 (1953), 51-64 (in Russian).
2. Khalatnikov I.M., Some questions of the relativistic hydrodynamics, J. Exp. Theor. Phys. 27 (1954), 529-541 (in Russian).
3. Landau L.D., Liftshitz E.M., Hydrodynamics, Nauka, Moscow, 1986 (in Russian).
4. Milekhin G.A., Nonlinear scalar fields and multiple particle production, Izv. AN SSSR Ser. Fiz. 26 (1962), 635-641 (in Russian).
5. Hwa R.C., Statistical description of hadron constituents as a basis for the fluid model of high-energy collisions, Phys. Rev. D 10 (1974), 2260-2268.
6. Cooper F., Frye G., Shonberg E., Landau's hydrodynamical model of particle production and electron-positron annihilation into hadrons, Phys. Rev. D 11 (1975), 192-213.
7. Chiu C.B., Sudarshan E.C.G., Wang K.-H., Hydrodynamical expansion with frame-independence symmetry in high-energy multiparticle production, Phys. Rev. D 12 (1975), 902-908.
8. Bjorken J.D., Highly relativistic nucleus-nucleus collisions: the central rapidity region, Phys. Rev. D 27 (1983), 140-151.
9. Csörgö T., Nagy M.I., Csanád M., New family of simple solutions of relativistic perfect fluid hydrodynamics, nucl-th/0605070.
10. Nagy M.I., Csörgö T., Csanád M., Detailed description of accelerating, simple solutions of relativistic perfect fluid hydrodynamics, arXiv:0709.3677.
11. Pratt S., A co-moving coordinate system for relativistic hydrodynamics, Phys. Rev. C 75 (2007), 024907, 14 pages, nucl-th/0612010.
12. Blandford R.D., McKee C.F., Fluid dynamics of relativistic blast waves, Phys. Fluids 19 (1975), 1130-1138.
13. Pan M., Sari R., Self-similar solutions for relativistic shocks emerging from stars with polytropic envelopes, Astrophys. Journ. 643 (2006), 416-422, astro-ph/0505176.
14. Zhdanov V.I., Borshch M.S., Ultra-relativistic expansion of ideal fluid with linear equation of state, J. Phys. Stud. 9 (2005), 233-237, hep-ph/0508220.
15. Korkina M.P., Martynenko V.G., Homogeneous symmetric model with extremely stiff equation of state, Ukraiïn. Fiz. Zh. 21 (1976), 1191-1196 (in Russian).
16. Scherrer R.J., Purely kinetic k-essence as unified dark matter, Phys. Rev. Lett. 93 (2004), 011301, 5 pages, astro-ph/0402316.
17. Chimento L.P., Lazkoz R., Atypical k-essence cosmologies, Phys. Rev. D 71 (2005), 023505, 16 pages, astro-ph/0404494.
18. Sahni V., Dark matter and dark energy, Lect. Notes Phys. 653 (2005), 141-180, astro-ph/0403324.
19. Copeland E.J., Sami M., Tsujikawa S., Dynamics of dark energy, Internat. J. Modern Phys. D 15 (2006), 1753-1936, hep-th/0603057.
20. Lichnerowicz A., Relativistic hydrodynamics and magnetodynamics, Benjamin, New York, 1967.
21. Gorenstein M.I., Zhdanov V.I., Sinyukov Yu.M., On scale-invariant solutions in the hydrodynamical theory of multiparticle production, J. Exp. Theor. Phys. 74 (1978), 833-845 (in Russian).
22. Gorenstein M.I., Sinyukov Yu.M., Zhdanov V.I., On scaling solutions in the hydrodynamical theory of multiparticle production, Phys. Lett. B 71 (1977), 199-202.
23. Grigoriev S.B., Korkina M.P., Nonlinear scalar fields and Friedmann models, in Gravitation and Electromagnetism, Editor F.I. Fedorov, Izd-vo Universitetskoje, Minsk, 1987, 14-21 (in Russian).
24. Babichev E., Mukhanov V., Vikman A., k-essence, superluminal propagation, causality and emergent geometry, arXiv:0708.0561.