### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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 Ostrovsky-Whitham 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
Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity 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; gradient-holonomic algorithm.

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