International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 10, Pages 487-534

Symmetry group analysis and invariant solutions of hydrodynamic-type systems

M. B. Sheftel1,2,3

1Department of Higher Mathematics, North-Western State Technical University, Millionnaya Street 5, St. Petersburg 191186, Russia
2Department of Physics, Boğaziçi University, Bebek, Istanbul 34342, Turkey
3Feza Gursey Institute, P.O. Box 6, Cengelkoy, Istanbul 81220, Turkey

Received 25 June 2002

Copyright © 2004 M. B. Sheftel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study point and higher symmetries of systems of the hydrodynamic type with and without an explicit dependence on t,x. We consider such systems which satisfy the existence conditions for an infinite-dimensional group of hydrodynamic symmetries which implies linearizing transformations for these systems. Under additional restrictions on the systems, we obtain recursion operators for symmetries and use them to construct infinite discrete sets of exact solutions of the studied equations. We find the interrelation between higher symmetries and recursion operators. Two-component systems are studied in more detail than n-component systems. As a special case, we consider Hamiltonian and semi-Hamiltonian systems of Tsarëv.