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**Hidden Symmetries of Lax Integrable Nonlinear Systems** ()

Recently devised new symplectic and
differential-algebraic approaches to studying hidden symmetry properties of
nonlinear dynamical systems on functional manifolds and their relationships to
Lax integrability are reviewed. A new symplectic approach to constructing
nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and
based upon the Marsden-Weinstein reduction method on canonically symplectic
manifolds with group symmetry, is described. Its natural relationship with the
well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated *R*-matrix method [1,2] is analyzed in detail. A new
modified differential-algebraic approach to analyzing the Lax integrability of
generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is
suggested and the corresponding Lax representations are constructed in exact
form. The related bi-Hamiltonian integrability and compatible Poissonian
structures of these generalized Riemann type hierarchies are discussed by means
of the symplectic, gradientholonomic and geometric methods.

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*Applied Mathematics*, Vol. 4 No. 10C, 2013, pp. 95-116. doi: 10.4236/am.2013.410A3013.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | L. D. Faddeev and L. A. Takhtadjan, “Hamiltonian Methods in the Theory of Solitons,” Springer, Berlin, 2000. |

[2] | A. G. Reyman and M. A. Semenov-Tyan-Shansky, “Reduction of Hamiltonian Systems, Affine Lie Algebras, and Lax Equations, I, II,” Invent. Math, Vol. 54, No. 1, 1979, pp. 81-100 and Vol. 63, No. 3, 1981, pp. 423-432. |

[3] | M. Blaszak, “Multi-Hamiltonian Theory of Dynamical Systems,” Springer, Berlin, 1998. |

[4] |
A. Newell, “Solitons in Mathematics and Physics,” SIAM, Philadelphia, 1985. http://dx.doi.org/10.1137/1.9781611970227 |

[5] | S. P. Novikov, “Theory of Solitons,” Springer, Berlin, 1984. |

[6] | A. G. Reyman and M. A. Semenov-Tian-Shansky, “Integrable Systems,” The Computer Research Institute Publishing, Moscow-Izhvek, 2003. (in Russian) |

[7] |
A. M. Mikhaylov, A. B. Shabat and R. I. Yamilov, “Extension of the Module of Invertible Transformations. Classification of Integrable Systems,” Communications in Mathematical Physics, Vol. 115, No. 1, 1988, pp. 1-19. http://dx.doi.org/10.1007/BF01238850 |

[8] | A. K. Prykarpatsky, O. D. Artemovych, Z. Popowicz and M. V. Pavlov, “Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 29, 2010, Article ID: 295205. |

[9] | Y. A. Prykarpatsky, O. D. Artemovych, M. Pavlov and A. K. Prykarpatsky, “The Differential-Algebraic and Bi-Hamiltonian Integrability Analysis of the Riemann Type Hierarchy Revisited,” Journal of Mathematical Physics, Vol. 53, 2012, Article ID: 103521. |

[10] | D. Blackmore, A. K. Prykarpatsky and V. Hr Samoylenko, “Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Differential-Geometrical Integrability Analysis,” World Scientific, New Jersey, 2011. |

[11] | Y. Mitropolsky, N. Bogolubov Jr., A. Prykarpatsky and V. Samoylenko, “Integrable Dynamical System: Spectral and Differential-Geometric Aspects,” Naukova Dunka, Kiev, 1987. (in Russian) |

[12] |
A. Prykarpatsky and I. Mykytyuk, “Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects,” Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. http://dx.doi.org/10.1007/978-94-011-4994-5 |

[13] | R. Abraham and J. E. Marsden, “Foundations of Mechanics,” Benjamin/Cummins Publisher, San Francisco, 1978. |

[14] |
V. I. Arnold, “Mathematical Methods of Classical Mechanics,” Springer, Berlin, 1989. http://dx.doi.org/10.1007/978-1-4757-2063-1 |

[15] |
M. Adler, “Completely Integrable Systems and Symplectic Action,” Journal of Mathematical Physics, Vol. 20, No. 1, 1979, pp. 60-67. http://dx.doi.org/10.1063/1.523963 |

[16] | A. M. Perelomov, “Integrable Systems of Classical Mechanics and Lie Algebras,” Nauka Publishing, Moscow, 1990. (in Russian) |

[17] | N. N. Bogolubov Jr. and Y. A. Prykarpatsky, “The Marsden-Weinstein Reduction Structure of Integrable Dynamical Systems and a Generalized Exactly Solvable Quantum Superradiance Model,” International Journal of Modern Physics B, Vol. 28, No. 1, 2012, pp. 237-245. |

[18] | R. V. Samulyak, “Generalized Dicke Type Dynamical System as the Inverse Nonlinear Schrodinger Equation,” Ukrainian Mathematical Journal, Vol. 47, No. 1, 1995, pp. 149-151. http://dx.doi.org/10.1007/BF01058807 |

[19] |
M. A. Semenov-Tian-Shansky, “What Is an R-Matrix?” Functional Analysis and Its Applications, Vol. 17, No. 4, 1983, pp. 259-272. http://dx.doi.org/10.1007/BF01076717 |

[20] | Y. A. Prykarpatsky, A. M. Samoilenko and A. K. Prykarpatsky, “The Geometric Properties of Canonically Reduced Symplectic Spaces with Symmetry, Their Relationship with Structures on Associated Principal Fiber Bundles and Some Applications,” Opuscula Mathematica, Vol. 25, No. 2, 2005, pp. 287-298. |

[21] | F. Calogero and A. Degasperis, “Spectral Transform and Solitons,” North-Holland, Amsterdam, 1982. |

[22] | J. Avan, O. Babelon and M. Talon, “Construction of Classical -Matrices for the Toda and Calogero Models,” Algebra and Analysis, Vol. 6, No. 2, 1994, p. 67. |

[23] | O. Babelon and C.-M. Viallet, “Hamiltonian Structures and Lax Equations,” Physics Letter B, Vol. 237, No. 3-4, 1990, pp. 411-416. |

[24] |
G. E. Arutyunov and P. B. Medvedev, “Generating Equation for -Matrices Related to the Dynamical Systems of Calogero Type,” Physics Letter A, Vol. 223, No. 1-2, 1996, pp. 66-74. http://dx.doi.org/10.1016/S0375-9601(96)00719-0 |

[25] | E. K. Sklyanin, “Quantum Variant of the Inverse Scattering Transform Method,” Proceedings of LOMI 95, Leningrad, 15-20 January 1980, pp. 55-128. (in Russian) |

[26] |
S. A. Tsyplyaev, “Commutation Relations for Transition Matrix in Classical and Quantum Inverse Scattering Method,” Theoretical and Mathematical Physics, Vol. 48, No. 1, 1981, pp. 24-33. (in Russian) http://dx.doi.org/10.1007/BF01037981 |

[27] | T. Crespo and Z. Hajto, “Algebraic Groups and Differential Galois Theory. Graduate Studies in Mathematics Series,” American Mathematical Society Publisher, Providence, 2011. |

[28] | I. Kaplanski, “Introduction to Differential Algebra,” Hermann, Paris, 1957. |

[29] | E. R. Kolchin, “Differential Algebra and Algebraic Groups,” Academic Press, New York, 1973. |

[30] | J. F. Ritt, “Differential Algebra,” AMS-Colloqium Publications, New York, 1966. |

[31] | J.-A. Weil, “Introduction to Differential Algebra and Differential Galois Theory,” CIMPA-UNESCO-Vietnam Lectures, Hanoi, 2001. |

[32] | J. Golenia, M. Pavlov, Z. Popowicz and A. Prykarpatsky, “On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogenious Regularizations and Their Integrability,” SIGMA 6, 2010, pp. 1-13. |

[33] | G. Wilson, “On the Quasi-Hamiltonian Formalism of the KdV Equation,” Physics Letter, Vol. 132, No. 8-9, 1988, pp. 445-450. |

[34] |
L. Brunelli and A. Das, “On an Integrable Hierarchy Derived from the Isentropic Gas Dynamics,” Journal of Mathematical Physics, Vol. 45, No. 7, 2004, p. 2633. http://dx.doi.org/10.1063/1.1756699 |

[35] | J. Golenia, N. N. Bogolubov Jr., Z. Popowicz, M. V. Pavlov and A. K. Prykarpatsky, “A New Riemann Type Hydrodynamical Hierarchy and Its Integrability Analysis,” 2009. http://publications.ictp.it |

[36] |
M. Pavlov, “The Gurevich-Zybin System,” Journal of Physics A: Mathematical and General, Vol. 38, No. 17, 2005, pp. 3823-3840. http://dx.doi.org/10.1088/0305-4470/38/17/008 |

[37] |
Z. Popowicz and A. K. Prykarpatsky, “The Non-Polynomial Conservation Laws and Integrability Analysis of Generalized Riemann Type Hydrodynamical Equations,” Nonlinearity, Vol. 23, No. 10, 2010, pp. 2517-2537. http://dx.doi.org/10.1088/0951-7715/23/10/010 |

[38] |
Y. Prykarpatsky, “Finite Dimensional Local and Nonlocal Reductions of One Type Hydrodynamic Systems,” Reports on Mathematical Physics, Vol. 50, No. 3, 2002, pp. 349-360. http://dx.doi.org/10.1016/S0034-4877(02)80065-9 |

[39] | A. K. Prykarpatsky and M. M. Prytula, “The GradientHolonomic Integrability Analysis of a Whitham-Type Nonlinear Dynamical Model for a Relaxing Medium with Spatial Memory,” Nonlinearity, Vol. 19, No. 9, 2006, pp. 2115-2122. http://dx.doi.org/10.1088/0951-7715/19/9/007 |

[40] |
J. P. Wang, “The Hunter-Saxton Equation: Remarkable Structures of Symmetries and Conserved Densities,” Nonlinearity, Vol. 23, No. 8, 2010, pp. 2009-2028. http://dx.doi.org/10.1088/0951-7715/23/8/011 |

[41] | Z. Popowicz, “The Matrix Lax Representation of the Generalized Riemann Equations and Its Conservation Laws,” Physics Letter A, Vol. 375, No. 37, 2011, pp. 3268-3272. |

[42] | J. C. Brunelli and S. Sakovich, “Hamiltonian Structures for the Ostrovsky-Vakhnenko Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 18, No. 1, 2013, pp. 56-62. |

[43] | A. Degasperis, D. D. Holm and A. N. W Hone, “A New Integrable Equation with Peakon Solutions,” Theoretical and Mathematical Physics, Vol. 133, No. 2, 2002, pp. 1463-1474. http://dx.doi.org/10.1023/A:1021186408422 |

[44] | L. A. Ostrovsky, “Nonlinear Internal Waves in a Rotating Ocean,” Okeanologia, Vol. 18, No. 2, 1978, pp. 181-191. |

[45] | A. A. Vakhnenko, “Solitons in a Nonlinear Model Medium,” Journal of Physics A, Vol. 25, No. 15, 1992, pp. 4181-4187. |

[46] | Y. Wang and Y. Chen, “Integrability of the Modified Generalized Vakhnenko Equation,” Journal of Mathematical Physics, Vol. 53, No. 12, 2012, Article ID: 123504. http://dx.doi.org/10.1063/1.4764845 |

[47] | G. B. Whitham, “Linear and Nonlinear Waves,” WileyInterscience, New York, 1974. |

[48] | O. Hentosh, M. Prytula and A. Prykarpatsky, “Differential-Geometric and Lie-Algebraic Foundations of Investigating Nonlinear Dynamical Systems on Functional Manifolds,” 2nd Edition, Lviv University Publishing, Lviv, 2006. (in Ukrainian) |

[49] | E. J. Parkes, “The Stability of Solutions of Vakhnenko’s Equation,” Journal of Physics A, Vol. 26, No. 22, 1993, pp. 6469-6475. |

[50] | A. G. Reyman and M. A. Semenov-Tian-Shansky, “The Hamiltonian Structure of Kadomtsev-Petviashvili Type Equations,” LOMI Proceedings, Leningrad, 12-17 January 1987, pp. 212-227. (in Russian) |

[51] | D. Blackmore, Y. A. Prykarpatsky, O. D. Artemowych, D. Orest and A. K. Prykarpatsky, “On the Complete Integrability of a One Generalized Riemann Type Hydrodynamic System,” arXiv:1204.0251v1. |

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