Аннотация:
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.
Ключевые слова:
generalized Riemann type hydrodynamical equations; Whitham typedynamical systems; Hamiltonian systems; Lax type integrability;gradient-holonomic algorithm.
Поступила:14 октября 2009 г.; в окончательном варианте 3 января 2010 г.; опубликована 7 января 2010 г.
Образец цитирования:
Jołanta Golenia, Maxim V. Pavlov, Ziemowit Popowicz, Anatoliy K. Prykarpatsky, “On a Nonlocal Ostrovsky–Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability”, SIGMA, 6 (2010), 002, 13 pp.
\RBibitem{GolPavPop10}
\by Jo\l anta Golenia, Maxim V.~Pavlov, Ziemowit Popowicz, Anatoliy K.~Prykarpatsky
\paper On a~Nonlocal Ostrovsky--Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability
\jour SIGMA
\yr 2010
\vol 6
\papernumber 002
\totalpages 13
\mathnet{http://mi.mathnet.ru/sigma459}
\crossref{https://doi.org/10.3842/SIGMA.2010.002}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2593380}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000273562500002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896059947}
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma459
https://www.mathnet.ru/rus/sigma/v6/p2
Эта публикация цитируется в следующих 21 статьяx:
Wei L., Wang Ya., “The Cauchy Problem For a Generalized Riemann-Type Hydrodynamical Equation”, J. Math. Phys., 62:4 (2021), 041502
Prykarpatskyy Ya.A., “Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin'S Integrability Classification of Perturbed Kdv-Type Equations”, Symmetry-Basel, 13:6 (2021), 1077
Wei L., “Wave Breaking, Global Existence and Persistent Decay For the Gurevich-Zybin System”, J. Math. Fluid Mech., 22:4 (2020), 47
Hentosh Oksana E, Balinsky Alexander A, Prykarpatski Anatolij K, “Poisson structures on (non)associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems”, Ann Math Phys, 3:1 (2020), 001
Orest Artemovych, Alexandr Balinsky, Anatolij Prykarpatski, “Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras”, ПМГЦ, 12:4 (2019)
Gao B., Tian K., Liu Q.P., Feng L., “Conservation Laws of the Generalized Riemann Equations”, J. Nonlinear Math. Phys., 25:1 (2018), 122–135
Samoilenko A.M., Prykarpatskyy Ya.A., Blackmore D., Prykarpatski A.K., “A Novel Integrability Analysis of a Generalized Riemann Type Hydrodynamic Hierarchy”, Miskolc Math. Notes, 19:1 (2018), 555–567
Artemovych O.D., Blackmore D., Prykarpatski A.K., “Poisson brackets, Novikov-Leibniz structures and integrable Riemann hydrodynamic systems”, J. Nonlinear Math. Phys., 24:1 (2017), 41–72
Tian K., Liu Q.P., “Conservation Laws and Symmetries of Hunter-Saxton Equation: Revisited”, Nonlinearity, 29:3 (2016), 737–755
Blackmore D., Prykarpatsky Ya.A., Bogolubov Jr. Nikolai N., Prykarpatski A.K., “Integrability of and Differential-Algebraic Structures for Spatially 1D Hydrodynamical Systems of Riemann Type”, Chaos Solitons Fractals, 59 (2014), 59–81
Prykarpatsky Ya.A., Artemovych O.D., Pavlov M.V., Prykarpatski A.K., “The Differential-Algebraic Analysis of Symplectic and Lax Structures Related with New Riemann-Type Hydrodynamic Systems”, Rep. Math. Phys., 71:3 (2013), 305–351
Prykarpatsky, Y.A., Blackmore, D., Golenia, J., Prykarpatsky, A.K., “A vertex operator representation of solutions to the gurevich-zybin hydrodynamical equation”, Opuscula Mathematica, 33:1 (2013), 139–149
Prykarpatsky Ya.A., Artemovych O.D., Pavlov M.V., Prykarpatsky A.K., “Differential-Algebraic and Bi-Hamiltonian Integrability Analysis of the Riemann Hierarchy Revisited”, J. Math. Phys., 53:10 (2012), 103521
Blackmore D., Prykarpatsky A.K., “The AKNS Hierarchy Revisited: a Vertex Operator Approach and its Lie-Algebraic Structure”, J. Nonlinear Math. Phys., 19:1 (2012), 1250001
Blackmore, D., Prykarpatsky, A.K., Prykarpatsky, Y.A., “Isospectral integrability analysis of dynamical systems on discrete manifolds”, Opuscula Mathematica, 32:1 (2012), 41–66
Prykarpatsky Ya.A., Bogolubov Nikolai N. Jr., Prykarpatsky A.K., Samoylenko V.H., “On the Complete Integrability of Nonlinear Dynamical Systems on Functional Manifolds Within the Gradient-Holonomic Approach”, Rep Math Phys, 68:3 (2011), 289–318
Popowicz Z., “The matrix Lax representation of the generalized Riemann equations and its conservation laws”, Phys Lett A, 375:37 (2011), 3268–3272
Pavlov M.V., Prykarpatsky A.K., “A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability”, Condensed Matter Physics, 13:4 (2010), 43002
Popowicz Z., Prykarpatsky A.K., “The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations”, Nonlinearity, 23:10 (2010), 2517–2537
Wang J.P., “The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities”, Nonlinearity, 23:8 (2010), 2009–2028
Цитирование в формате AMSBIB
Обратная связь: