Isospectral integrability analysis of dynamical systems on discrete manifolds

• Autorzy: Blackmore D. , Prykarpatsky A. K. , Prykarpatsky Y. A.
• Języki publikacji: EN

Abstrakty

EN

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems.

Słowa kluczowe

EN

gradient holonomic algorithm; conservation laws; asymptotic analysis; Poissonian structures; Lax representation; finite dimensional reduction; Liouville integrability; nonlinear discrete dynamical systems

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 1
• Strony: 41--66
• Opis fizyczny: Bibliogr. 37 poz.

Twórcy

autor

Blackmore D.

autor

Prykarpatsky A. K.

autor

Prykarpatsky Y. A.

• AGH University of Science and Technology Faculty of Mining Surveying and Environmental Engineering al. Mickiewicza, 30-059 Krakow, Poland,

Bibliografia

• [1] R. Abraham, J. Marsden, Foundation of mechanics, The Benjamin/Cummings Publ. C., Masachusets, 1978.
• [2] V.I. Arnold, Mathematical methods of classical mechanics, Springer, New York, 1978.
• [3] P.D. Lax, Periodic solutions of the KdV equation, Commun. Pure and Appl. Math. 28 (1975) 421, 141–188.
• [4] S.P. Novikov (Ed.), Theory of Solitons, Moscow, Nauka, 1980 [in Russian].
• [5] M. Ablowitz, J. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16 (1975) 3, 598–603.
• [6] A. Prykarpatsky, I. Mykytyuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Kluwer Academic Publishers, the Netherlands, 1998.
• [7] O. Hentosh, M. Prytula, A. Prykarpatsky, Differential-geometric and Lie-algebraic foundations of investigating nonlinear dynamical systems on functional manifolds, 2nd ed., Lviv University Publication, Lviv, 2006 [in Ukrainian].
• [8] M. Błaszak, Multi-Hamiltonian theory of dynamical systems, Springer, Berlin-Heidelberg, 1998.
• [9] R.K. Dodd, J.C. Eilbeck, J. Gibbon, H.C. Morris, Solitons and nonlinear equations, Academic Press, London, 1984.
• [10] F. Magri, A geometric approach to the nonlinear solvable equations, Lecture Notes in Phys. (1980) 120, 233–263.
• [11] Yu.A. Mitropolsky, V.H. Samoylenko, Differential-difference dynamical systems, associtated with the difference dirac operator, and their complete integrability, Ukrainian Math. J. 37 (1985) 2, 180–186 [in Russian].
• [12] A.K. Prykarpatsky, Elements of the integrability theory of discrete dynamical systems, Ukrainian Math. J. 39 (1987) 1, 298–301 [in Russian].
• [13] D. Blackmore, A.K. Prykarpatsky, V. Samoylenko, Nonlinear dynamical systems of mathematical physics: spectral and differential-geometric integrability analysis, World Scientific, New York, 2011.
• [14] N.N. Bogolubov (Jr.), A.K. Prykarpatsky, Inverse periodic problem for a discrete approximation of the nonlinear Schrödinger equation, Dokl. AN SSSR 265 (1982) 5, 1103–1108.
• [15] Yu.A. Mitropolsky, N.N. Bogolubov (Jr.), A.K. Prykarpatsky, V.H. Samoylenko, Integrable dynamical systems: differential-geometric and spectral aspects, Naukova Dumka, Kyiv, 1987 [in Russian].
• [16] O.I. Bogoyavlensky, S.P. Novikov, On the connection of Hamiltonian formulations of stationary and nonstationary problems, Funct. Anal. Appl. 10 (1976) 1, 9–13 [in Russian].
• [17] I.M. Gelfand, L.A. Dikiy, Integrable nonlinear equations and the Liouville theorem, Funct. Anal. Appl. 13 (1979) 1, 8–20 [in Russian].
• [18] B. Fuchsteiner, A. Fokas, Symplectic structures, Backlund transformatrions and hereditary symmetries, Phys. D 4 (1981) 1, 47–66.
• [19] B.A. Kupershmidt, Discrete Lax equations and differential-difference calculus, Astérisque 123 (1985), 5–212.
• [20] P. Deift, L.-C. Li, C. Tomei, Loop groups, discrete versions of some classical integrable systems and rank-2 extensions, Mem. Amer. Math. Soc. 100 (1992) 479, 1–101.
• [21] J. Moser, A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, ETH, Zurich, 1989, Preprint, 76 pp.
• [22] J.C. Baez, J.W. Gillam, An algebraic approach to discrete mechanics, Lett. Math. Phys. 31 (1994) 3, 205–212.
• [23] M. Błaszak, A. Szum, A. Prykarpatsky, Central extension approach to integrable field and lattice-field systems in (2+1)-dimensions, Rep. Math. Phys. 37 (1999) 5, 27–32.
• [24] O. Ragnisco, G.Z. Tu, A new hierarchy of integrable discrete systems, Preprint of Dept. of Physics University of Roma, 1989.
• [25] A.C. Newell, Solitons in mathematics and physics, SIAM Publ., Arizona, 1985.
• [26] L.D. Faddeev, L.A. Takhtadjan, Hamiltonian methods in the theory of solitons, Springer, New York, 1987.
• [27] A.G. Reyman, M.A. Semenov-Tian-Shansky, Integrable systems, R&C-Dynamics, Moscow-Izhevsk, 2003 [in Russian].
• [28] M.A. Semenov-Tian-Shansky, What is the classical R-matrix?, Funct. Anal. Appl. 17 (1983) 4, 17–33 [in Russian].
• [29] M. Jimbo, T. Miwa, Solutions and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943–10001.
• [30] B.A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and Integrable systems, Amer. Math. Soc. 78 (2000).
• [31] A.K. Prykarpatsky, O.D. Artemovych, Z. Popowicz, M.V. Pavlov, Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg–de Vries hydrodynamical equations, J. Phys. A: Math. Theor. 43 (2010) 295205 (13 pp).
• [32] J. Golenia, M. Pavlov, Z. Popowicz, A. Prykarpatsky, On a nonlocal Ostrovsky-Whitham type dynamical system, its Riemann type inhomogenious regularizations and their integrability, SIGMA 6 (2010) 2, 1–13.
• [33] 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, 2009.
• [34] M. Pavlov, The Gurevich-Zybin system, J. Phys. A: Math. Gen. 38 (2005) 3823, 3823–3840.
• [35] O. Hentosh, A. Prykarpatsky, V.H. Samoylenko, The Lie-algebraic structure of Lax type integrable nonlocal differential-difference equations, Journal of Nonlinear Oscillations 3 (2000) 1, 84–94.
• [36] C. Hardouin, M.F. Singer, Differential Galois theory of linear difference equations, Math. Ann. 342 (2008) 2, 333–377.
• [37] G. Wilson, On the quasi-Hamiltonian formalism of the KdV equation, Phys. Lett. 132 (1988) 8–9, 445–450