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A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

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Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type hydrodynamical hierarchy is devised. A functional representation generating an infinite hierarchy of dispersive Lax type integrable flows is obtained.
Rocznik
Strony
139--149
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
autor
  • University of Agriculture, Department of Applied Mathematics, Balicka 253c, 30-198 Kraków, Poland, yarpry@gmail.com
Bibliografia
  • [1] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974, 221p.
  • [2] A.V. Gurevich, K.P. Zybin, Nondissipative gravitational turbulence, Sov. Phys.-JETP, 67 (1988), 1-12.
  • [3] A.V. Gurevich, K.P. Zybin, Large-scale structure of the Universe, Analytic Theory Sov. Phys. Usp. 38 (1995), 687-722.
  • [4] J. Golenia, N. Bogolubov (Jr.), Z. Popowicz, M. Pavlov, A. Prykarpatsky, A new Riemann type hydrodynamical hierarchy and its integrability analysis, Preprint ICTP, IC/2009/095, 2010.
  • [5] J. Golenia, M. Pavlov, Z. Popowicz, A. Prykarpatsky, On a nonlocal Ostrovsky-Whitham type dynamical system, its Riemann type inhomogenious regularizations and their inte grability, SIGMA 6 (2010), 1-13.
  • [6] A.K. Prykarpatsky, M.M. Prytula, The gradient-holonomic integrability analysis of a Whitham-type nonlinear dynamical model for a relaxing medium with spatial memory, Nonlinearity 19 (2006), 2115-2122.
  • [7] Z. Popowicz, The matrix Lax representation of the generalized Riemann equations and its conservation laws, Physics Letters. A 375 (2011), 3268-3272.
  • [8] Z. Popowicz, A.K. Prykarpatsky, The non-polynomial conservation laws and integra bility analysis of generalized Riemann type hydrodynamical equations, Nonlinearity 23 (2010), 2517-2537.
  • [9] M. Pavlov, The Gurevich-Zybin system, J. Phys. A: Math. Gen. 38 (2005), 3823-3840.
  • [10] S.P. Novikov (ed.), Theory of Solitons, Nauka Publ., Moscow, 1980 [in Russian],
  • [11] V.E. Zakharov, S.M. Manakov, Construction of multi-dimensional nonlinear integrable systems and their solutions, Funct. Anal. Appl. 19 (1985) 2, 11-25 [in Russian],
  • [12] S.P. Burtsev, V.E. Zakharov, A.V. Mikhaylov, Inverse scattering problem with changing spectral parameter, Theor. Math. Phys. 70 (1987) 3, 323-341 [in Russian].
  • [13] S.M. Manakov, P.M. Santini, On the solutions of the second heavenly and Pavlov equa tions, J.Phys. A: Math and Theor. 42 (2009) 40, 404013-10.
  • [14] A.C. Newell, Solitons in mathematics and physics, SIAM Publ., Arizona, 1985.
  • [15] L.A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, NY, 1991.
  • [16] L.D. Faddeev, L.A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons, Springer, NY, 1987.
  • [17] A. Prykarpatsky, I. Mykytyuk, Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, Kluwer Academic Publishers, the Netherlands, 1998, 553p.
  • [18] D. Blackmore, A.K. Prykarpatsky, V.H. Samoylenko, Nonlinear Dynamical Systems of Mathematical Physics, World Scientific, NY, 2011.
  • [19] O.Ye. Hentosh, M.M. Prytula, A.K. Prykarpatsky, Differential-geometric integrability fundamentals of nonlinear dynamical systems on functional menifolds, (The second revised edition), Lviv University Publisher, Lviv, Ukraine, 2006 [in Ukrainian],
  • [20] Yu.A. Mitropolski, N.N. Bogoliubov (Jr.), A.K. Prykarpatsky, V.Hr. Samoilenko, Integrable Dynamical Systems, Nauka dumka, Kiev, 1987 [in Russian],
  • [21] M. Blaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer, Berlin, 1998.
  • [22] G.M. Pritula, V.E. Vekslerchik, Conservation laws for the nonlinear Schrodinger equa tion in Miwa variables, Inverse Problems 18 (2002), 1355.
  • [23] D. Blackmore, A.K. Prykarpatsky, Y.A. Prykarpatsky, Iso-spectrally integrable dynami cal systems on discrete manifolds: analytical aspect, Opuscula Math. 32 (2012) 1, 41-65.
  • [24] D. Blackmore, A.K. Prykarpatsky, The AKNS hierarchy revisited: A vertex operator approach and its Lie-algebraic structure, J. Nonlinear Mathematical Phys. 19 (2012) 1. 1250001, pp. 15.
  • [25] A.K. Prykarpatsky, O.D. Artemovych, Z. Popowicz, M.V. Pavlov, Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrody-namical equations, J. Phys. A: Math. Theor. 43 (2010), 295205-15.
  • [26] A.G. Reyman, Semenov-Tian-Shansky, Integrable Systems, R&C-Dynamics, Moscow-Izhevsk, 2003 [in Russian]
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0009-0013
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