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On the integrability of a new generalized Gurevich-Zybin dynamical system, its Hunter-Saxton type reduction and related mysterious symmetries

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There is studied the integrability of a generalized Gurevich-Zybin dynamical system based on the differential-algebraic and geometrically motivated gradient-holonomic approaches. There is constructed the corresponding Lax type represenation, compatible Poisson structures as well as the integrability of the related Hunter-Saxton reduction. In particular, there are constructed its Lax type repreentation, the Hamiltonian symmetries as flows on a functional manifold endowed with compatible Poisson structures as well as so called new mysterious symmetries, depending on functional parameter. Similar results are also presented for the potential-KdVdynamical system, for which we also obtained its new mysterious symmetries first presented in a clear, enough short and analytically readable form.

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References

  1. Ferapontov, E.V.: Reciprocal transformations and their invariants. Differ. Equ. 25(7), 898–905 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Ferapontov, E.V., Rogers, C., Schief, W.K.: Reciprocal transformations of two-component hyperbolic systems and their invariants. J. Math. Anal. Appl. 228(2), 365–376 (1998)

    Article  MathSciNet  Google Scholar 

  3. Miura, R.: Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation, J. Math. Phys. 9: 1202–1204 (1968); https://doi.org/10.1063/1.1664700

  4. Rogers, C.: Reciprocal transformations in (2 + 1) dimensions. J. Phys. A Math. Gen. 19, L491–L496 (1986)

    Article  MathSciNet  Google Scholar 

  5. Rozhdestvensky, K.: Computer Modeling and Simulation Dynamic Systems Using Wolfram System Modeler. Springer, Berlin (2020)

    Book  Google Scholar 

  6. Mitropolski Yu.A., Bogoliubov N.N. (Jr.), Prykarpatsky A.K., Samoilenko V.Hr. Integrable Dynamical Systems. "Naukova dumka’ Publ., Kiev, 1987 (in Russian)

  7. Blackmore, D., Prykarpatsky, A.K., Samoylenko, V.H.: Nonlinear dynamical systems of mathematical physics. World Scientific Publisher, NJ, USA (2011)

    Book  Google Scholar 

  8. Hentosh, O., Prytula, M., Prykarpatsky, A.: Differential-geometric and Li-algebraic foundations of investigating integrable nonlinear dynamical systems on functional manifolds.—Lviv. University, 2006.—408 p. (in Ukrainian)

  9. Prykarpatsky, A.K., Mykytiuk, I.V.: Algebraic integrability of nonlinear dynamical systems on manifolds. - Kluwer, 553 (1998)

  10. Dryuma, V.S.: On the Riemann and Einstein–Weyl geometry in theory of the second order ordinary differential equations. Bul. Acad. Sti. Rep. Moldova (Phys., Tech.), 3, 95–102 (1999). arXiv:gr-qc/0104095

  11. Tod, K.P.: Einstein-Weyl spaces and third order differential equations. J. Math. Phys. 41, 5572–5581 (2000)

    Article  MathSciNet  Google Scholar 

  12. Golovin, S.V. :Group foliation of Euler equations in nonstationary rotationally symmetrical case. In: Proc. Inst. Math. of NAS of Ukraine, vol. 50, no. Part 1, (2004), 110–117

  13. Morozov, O.I.: Integrability structures of the generalized Hunter-Saxton equation. Anal. Math. Phys. 11(50), 1–21 (2021). https://doi.org/10.1007/s13324-021-00490-8

    Article  MathSciNet  MATH  Google Scholar 

  14. Morozov, O.I.: Structure of symmetry groups via Cartan’s method: survey of four approaches. Symmetry, Integrability and Geometry: Methods and Applications, 1, Paper 006, 14 (2005)

  15. Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal symmetries and the theory of coverings. Acta Appl. Math., pp. 79–86 (1984)

  16. Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math., pp. 161–209 (1989)

  17. Vinogradov, A.M., Krasil’shchik, I.S. (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics [in Russian], Moscow: Factorial, 2005; English transl. prev. ed.: I.S. Krasil’shchik, A.M. Vinogradov (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Transl. Math. Monogr., 182, Amer. Math. Soc., Providence, RI (1999)

  18. Gurevich A.V., Zybin, K.P.: Nondissipative gravitational turbulence Sov. Phys.–JETP, 67(1988), 1–12

  19. Gurevich, A.V., Zybin, K.P.: Large-scale structure of the Universe. Analytic Theory Sov. Phys. Usp. 38, 687–722 (1995)

    Google Scholar 

  20. Pavlov, M.V.: The Gurevich-Zybin system. J. Phys. A: Math. Gen. 38, 3823–3840 (2005). https://doi.org/10.1088/0305-4470/38/17/008

    Article  MathSciNet  MATH  Google Scholar 

  21. Beals, R., Sattinger, D.H., Szmigielski, J.: Inverse scattering solutions of the Hunter-Saxton equation. Appl. Anal. N 3–4, 255–269 (2001)

    Article  MathSciNet  Google Scholar 

  22. Morozov, O.I.: Contact equivalence of the generalized Hunter-Saxton and the Euler-Poisson equation. 2004. Preprint arXiv:math-ph/0406016

  23. Pavlov, M.V., Prykarpatsky, A.K.: A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability. Condensed Matter Physics, 2010, Vol. 13, No 4, 43002: 1–21; https://www.icmp.lviv.ua/journal

  24. Prykarpatsky, Y.A., Blackmore, D., Golenia , J., Prykarpatsky, A.K.: A vertex operator representation of solutions to teh Gurevich-Zybin hydropdynamical equation. Opuscula Math. 33, no. 1, 139–149 (2013) https://doi.org/10.7494/OpMath.2013.33.l.139

  25. Prykarpatsky, Y.A., Artemovych, O.D., Pavlov, M.V., Prykarpatsky, A.K.: The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hydrodynamic systems. Reports on Math. Phys., Vol. 71/3 (2013), 305–351

  26. Prykarpatsky, Y.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. (2012). https://doi.org/10.1063/1.4761821

    Article  MathSciNet  MATH  Google Scholar 

  27. Prykarpatsky, A., Prytula, M.: The gradient-holonomic integrability analysis of a Witham-type nonlinear dynamical model for a relaxing medium with spatial memory. Nonlinearity, 19, 2115–2122 (2006) https://doi.org/10.1088/0951-7715/19/9/007

  28. Tian, K., Liu, Q.P.: Conservation laws and symmetries of Hunter-Saxton equation revisited. Nonlinearity 29(3), 1–5 (2015). https://doi.org/10.1088/0951-7715/29/3/737

    Article  MathSciNet  Google Scholar 

  29. Reyes, E.G.: The soliton content of the Camassa–Holm and Hunter–Saxton equations. In: A.G. Nikitin, A.G. Boyko, Popovych, R.O. (eds.) Proceedings of the Fourth International Conference on Symmetry in Nonlinear Mathematical Physics/Proceedings of the Institute of Mathematics of the NAS of Ukraine, vol. 43. Kyiv, pp. 201–208 (2002)

  30. Wang, J.P.: The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities. Nonlinearity 23, 2009–2028 (2010)

    Article  MathSciNet  Google Scholar 

  31. Hunter, J.K., Saxton, R.: Dynamics of director elds. SIAM J. Appl. Math. 51(1498), 1521 (1991)

    MATH  Google Scholar 

  32. Prykarpatsky, A.K., Artemovych, O.D., Popowicz Z., Pavlov, M.V.: Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg—de Vries hydrodynamical equations, J. Phys. A, Math. Theor. 43, 295205 (2010)

  33. Faddeev, L.D., Takhtadjian, L.A.: Hamiltonian Approach in Soliton Theory. Springer, Berlin-Heidelberg (1987)

    Book  Google Scholar 

  34. Krasil’shchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of jet spaces and nonlinear partial differential equations. Gordon and Breach, New York (1986)

    MATH  Google Scholar 

  35. Novikov S.P. (Editor) Theory of Solitons, Springer, (1984)

  36. Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl. 8(3), 236–246 (1974)

    Article  MathSciNet  Google Scholar 

  37. Fokas, A., Fuchssteiner, B.: Symplectic structures, their Backlund transformations and hereditary symmetries. Physica 4D, 47 (1981)

    MATH  Google Scholar 

  38. Olver, P.J.: Applications of Lie groups to differential equations, 2 ed.; Vol. 107, Graduate Texts in Mathematics, Springer-Verlag: New York, 1993; p. 500; https://doi.org/10.1007/978-1-4684-0274-2.

  39. Blackmore, D., Prykarpatsky, A.K.: Dark equations and their light integrability. J. Nonlinear Math. Phys. 21(3), 407–428 (2014). https://doi.org/10.1080/14029251.2014.936760

    Article  MathSciNet  MATH  Google Scholar 

  40. Blaszak, M.: Multi-Hamiltonian Theory of Dynamical Systems. Springer, Berlin (1998)

    Book  Google Scholar 

  41. Rasin, A., Schiff, J.: Four Symmetries of the KdV equation. arXiv:2108.01550v1 [nlin.SI] 3 Aug 2021

  42. : Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. AMS, Mathematical Surveys and Monographs, Vol. 53, (1997)

  43. Abraham, R., Marsden, J.: Foundations of Mechanics, 2nd edn. Benjamin Cummings, NY (1978)

    MATH  Google Scholar 

  44. Arnold, V.I.: Mathematical methods of classical mechanics. Springer, Berlin (1989)

    Book  Google Scholar 

  45. Hunter, J.K., Zheng, Y.: On a completely integrable nonlinear hyperbolic variational equation. Physica D 79, 361–386 (1994)

    Article  MathSciNet  Google Scholar 

  46. Golenia, J., Pavlov, M., Popowicz, R., Prykarpatsky, A.: On a nonlocal Ostrovsky-Witham type dynamical system, its Riemann type inhomogenious regularizations and their integrability. SIGMA 6, 1–13 (2010)

    MATH  Google Scholar 

  47. Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)

    Article  MathSciNet  Google Scholar 

  48. Kaup, D.J.: A higher-order water-wave equation and the method for solving it. Prog. Theor. Phys. 54, 396–408 (1975)

    Article  MathSciNet  Google Scholar 

  49. Popowicz, Z., Prykarpatsky, A.: The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations. Nonlinearity 23, 2517–2537 (2010)

    Article  MathSciNet  Google Scholar 

  50. Prykarpatsky, Y.A.: Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations. Symmetry 13, 1077 (2021). https://doi.org/10.3390/sym130610

    Article  Google Scholar 

  51. Samoilenko, A.M., Prykarpatsky, Ya.A.: Algebraic-analytical aspects of completely integrable dynamical systems and their perturbations.—Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine, 2002. 237

  52. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)

    Article  MathSciNet  Google Scholar 

  53. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)

    Article  MathSciNet  Google Scholar 

  54. Fil, B.M., Prykarpatski, A.K., Prutula, N.N.: Quantum Lie algebra of currents—the universal algebraic structure of symmetries of completely integrable dynamical systems. Ukrainian Math. J. 40, 645–649 (1988)

    Article  MathSciNet  Google Scholar 

  55. Bogolyubov N.N. (Jr.)., Prykarpatskii, A.K.: Quantum current Lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics. Theoretical and Mathematical Physics, 1988, 329–339

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Acknowledgements

The authors are indebted to our colleagues Alexander Balinsky and Radoslaw Kycia with whom discussions during preparation of the manuscript were very useful. The last but not least appreciation belongs to Referee for important suggestions and remarks which proved very instrumental when preparing a manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences at the NJIT, Newark, NJ, USA

    Denis Blackmore

  2. Department of Applied Mathematics at the Cracov Agriculture University, Krakow, Poland

    Yarema Prykarpatsky

  3. The Ivan Franko National University of Lviv, Lviv, Ukraine

    Mykola M. Prytula

  4. LAMA/CNRS UMR 5127, Laboratoire de Mathematiques, Universite Savoie Mont Blanc, Campus Scientifique, 73376, Le Bourget-du-Lac, France

    Denys Dutykh

  5. Cracow University of Technology, Krakow, Poland

    Anatolij K. Prykarpatski

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  1. Denis Blackmore
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Blackmore, D., Prykarpatsky, Y., Prytula, M.M. et al. On the integrability of a new generalized Gurevich-Zybin dynamical system, its Hunter-Saxton type reduction and related mysterious symmetries. Anal.Math.Phys. 12, 66 (2022). https://doi.org/10.1007/s13324-022-00662-0

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