
SIGMA 6 (2010), 002, 13 pages arXiv:1001.1145
https://doi.org/10.3842/SIGMA.2010.002
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics
On a Nonlocal OstrovskyWhitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability
Jolanta Golenia ^{a}, Maxim V. Pavlov ^{b}, Ziemowit Popowicz ^{c} and Anatoliy K. Prykarpatsky ^{d, e}
^{a)} The Department of Applied Mathematics, AGH
University of Science and Technology, Kraków 30059, Poland
^{b)} Department of Mathematical Physics, P.N. Lebedev Physical Institute,
53 Leninskij Prospekt, Moscow 119991, Russia
^{c)} The Institute for Theoretical Physics, University of Wroclaw, Wroclaw 50204, Poland
^{d)} The Department of Mining Geodesics, AGH University
of Science and Technology, Kraków 30059, Poland
^{e)} Department of Economical Cybernetics, Ivan Franko State Pedagogical University,
Drohobych, Lviv Region, Ukraine
Received October 14, 2009, in final form January 03, 2010; Published online January 07, 2010
Abstract
Shortwave perturbations in a relaxing medium, governed by a special
reduction of the Ostrovsky evolution equation, and later derived by Whitham,
are studied using the gradientholonomic integrability algorithm. The
biHamiltonicity and complete integrability of the corresponding dynamical
system is stated and an infinite hierarchy of commuting to each other
conservation laws of dispersive type are found. The well defined
regularization of the model is constructed and its Lax type integrability is
discussed. A generalized hydrodynamical Riemann type system is considered,
infinite hierarchies of conservation laws, related compatible Poisson
structures and a Lax type representation for the special case N=3 are
constructed.
Key words:
generalized Riemann type hydrodynamical equations; Whitham type dynamical systems; Hamiltonian systems; Lax type integrability; gradientholonomic algorithm.
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