International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 10, Pages 487-534
doi:10.1155/S0161171204206147
Symmetry group analysis and invariant solutions of hydrodynamic-type systems
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.
Abstract
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.