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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References

1. Blaszak M., Multi-Hamiltonian theory of dynamical systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1998.
2. Bogolyubov N.N. Jr., Prykarpatsky A.K., Gucwa I., Golenia J., Analytical properties of an Ostrovsky-Whitham type dynamical system for a relaxing medium with spatial memory and its integrable regularization, Preprint IC/2007/109, Trieste, Italy, 2007, arXiv:0902.4395.
3. Bogolyubov N.N. Jr., Golenia J., Popowicz Z., Pavlov M.V., Prykarpatsky A.K., A new Riemann type hydrodynamical hierarchy and its integrability analysis, Preprint IC/2009/095, Trieste, Italy, 2009.
4. Brunelli J.C., Das A., On an integrable hierarchy derived from the isentropic gas dynamics, J. Math. Phys. 45 (2004), 2633-2645, nlin.SI/0401009.
5. Chorin A.J., Marsden J.E., A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, Vol. 4, Springer-Verlag, New York, 1993.
6. Davidson R.C., Methods in nonlinear plasma theory, Academic Press, New York, 1972.
7. Faddeev L.D., Takhtajian L.A., Hamiltonian methods in the theory of solitons, Classics in Mathematics, Springer, Berlin, 2007.
8. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
9. Gurevich A.V., Zybin K.P., Nondissipative gravitational turbulence, Soviet Phys. JETP 67 (1988), 1-12.
10. Gurevich A.V., Zybin K.P., Large-scale structure of the Universe. Analytic theory, Soviet Phys. Usp. 38 (1995), 687-722.
11. Hentosh O., Prytula M., Prykarpatsky A., Differential-geometric and Lie-algebraic foundations of investigating nonlinear dynamical systems on functional manifolds, 2nd ed., Lviv University Publ., 2006 (in Ukrainian).
12. Hunter J.K., Saxton R., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521.
13. Hunter J.K., Zheng Y.X., On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994), 361-386.
14. Lenells J., The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal. 40 (2008), 266-277.
15. Magri F., A simple model of the integrable Hamiltonian equations, J. Math. Phys. 19 (1978), 1156-1162.
16. Mitropol'skij Yu.A., Bogolyubov N.N. Jr., Prikarpatskij A.K., Samojlenko V.G., Integrable dynamical systems: spectral and differential-geometric aspects, Naukova Dumka, Kiev, 1987 (in Russian).
17. Morrison A.J., Parkes E.J., Vakhnenko V.O., The N loop soliton solution of the Vakhnenko equation, Nonlinearity 12 (1999), 1427-1437.
18. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
19. Olver P.J., Rosenau P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996), 1900-1906.
20. Ostrovsky L.A., Nonlinear internal waves in a rotating ocean, Oceanology 18 (1978), 119-125.
21. Parkes E.J., The stability of solution of Vakhnenko's equation, J. Phys. A: Math. Gen. 26 (1993), 6469-6475.
22. Parkes E.J., Vakhnenko V.O., Explicit solutions of the Camassa-Holm equation, Chaos Solitons Fractals 26 (2005), 1309-1316.
23. Pavlov M.V., The Gurevich-Zybin system, J. Phys. A: Math. Gen. 38 (2005), 3823-3840, nlin.SI/0412072.
24. Prykarpatsky A.K., Mykytyuk I.V., Algebraic integrability of nonlinear dynamical systems on manifolds. Classical and quantum aspects, Mathematics and its Applications, Vol. 443, Kluwer Academic Publishers Group, Dordrecht, 1998.
25. Prykarpatsky A.K., Prytula M.M., The gradient-holonomic integrability analysis of a Whitham-type nonlinear dynamical model for a relaxing medium with spatial memory, Nonlinearity 19 (2006), 2115-2122.
26. Prykarpatsky A.K., Prytula M.M., The gradient-holonomic analysis of the integrability of a nonlinear Whitham-type model for a relaxing medium with memory, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2006), no. 5, 13-18 (in Ukrainian).
27. Sakovich S., On a Whitham-type equation, SIGMA 5 (2009), 101, 7 pages, arXiv:0909.4455.
28. Whitham G.B., Linear and nonlinear waves, Wiley-Interscience, New York - London - Sydney, 1974.