July  2015, 35(7): 2949-2977. doi: 10.3934/dcds.2015.35.2949

Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions

1. 

ZHAW School of Engineering, Technikumstrasse 9, CH-8401 Winterthur, Switzerland

Received  March 2014 Revised  December 2014 Published  January 2015

Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.
Citation: Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949
References:
[1]

V. I. Arnol'd, V. V. Kozlov and A. I. Neishtadt, Dynamical Systems III (Mathematical Aspects of Classical and Celestial Mechanics), Third edition, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.

[2]

R. F. Bikbaev and S. B. Kuksin, On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. Krichever, Lett. Math. Phys., 28 (1993), 115-122. doi: 10.1007/BF00750304.

[3]

E. Date and S. Tanaka, Analogue of inverse scattering theory for the discrete Hill's equation and exact solutions for the periodic Toda lattice, Progr. Theor. Phys., 55 (1976), 457-465. doi: 10.1143/PTP.55.457.

[4]

H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. Sect. B, 9 (1974), 1924-1925. doi: 10.1103/PhysRevB.9.1924.

[5]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, in Collected Papers of Enrico Fermi, University of Chicago Press, Chicago, 2 (1965), 978-988; Theory, Methods and Applications, 2nd edition, Marcel Dekker, New York, 2000.

[6]

B. Grébert and T. Kappeler, Symmetries of the nonlinear Schrödinger equation, Bull. Soc. math. France, 130 (2002), 603-618.

[7]

A. Henrici and T. Kappeler, Global action-angle variables for the periodic Toda lattice, Int. Math. Res. Not., (2008), Art. ID rnn031, 52 pp. doi: 10.1093/imrn/rnn031.

[8]

A. Henrici and T. Kappeler, Global Birkhoff coordinates for the periodic Toda lattice, Nonlinearity, 21 (2008), 2731-2758. doi: 10.1088/0951-7715/21/12/001.

[9]

A. Henrici and T. Kappeler, Birkhoff normal form for the periodic Toda lattice, in Integrable Systems and Random Matrices, Contemp. Math., 458, American Mathematical Society, 2008, Providence, RI, 11-19.

[10]

A. Henrici and T. Kappeler, Results on normal forms for FPU chains, Comm. Math. Phys., 278 (2008), 145-177. doi: 10.1007/s00220-007-0387-z.

[11]

A. Henrici and T. Kappeler, Resonant normal form for even periodic FPU chains, J. Eur. Math. Soc., 11 (2009), 1025-1056. doi: 10.4171/JEMS/174.

[12]

A. Henrici and T. Kappeler, Nekhoroshev theorem for the periodic Toda lattice, Chaos, 19 (2009), 033120, 13 pp. doi: 10.1063/1.3196783.

[13]

T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation, Comm. Math. Phys., 285 (2009), 1087-1107. doi: 10.1007/s00220-008-0543-0.

[14]

T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[15]

T. Kappeler and P. Topalov, Global Well-Posedness of KdV in $H^{-1}(\mathbbT,\mathbbR)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215/S0012-7094-06-13524-X.

[16]

P. Lochak, Hamiltonian perturbation theory: Periodic orbits, resonances and intermittency, Nonlinearity, 6 (1993), 885-904. doi: 10.1088/0951-7715/6/6/003.

[17]

P. Lochak and A. Neishtadt, Estimates of stability time for nearly integrable systems with a quasi-convex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.

[18]

P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81. doi: 10.1007/BF01418827.

[19]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I, Uspekhi Mat. Nauk, 32 (1977), 5-66; Russian Math. Surveys, 32 (1977), 1-65.

[20]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems II, Trudy Sem. Petrovsk., 5 (1979), 5-50.

[21]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.

[22]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.

[23]

J. Pöschel, On Nekhoroshev's estimate at an elliptic equilibrium, Int. Math. Res. Not., 4 (1999), 203-215. doi: 10.1155/S1073792899000100.

[24]

B. Rink, Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice, Physica D, 175 (2003), 31-42. doi: 10.1016/S0167-2789(02)00694-2.

[25]

B. Rink, Proof of Nishida's conjecture on anharmonic lattices, Comm. Math. Phys., 261 (2006), 613-627. doi: 10.1007/s00220-005-1451-1.

[26]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys and Monographs, 72, Amer. Math. Soc., Providence, 2000.

[27]

M. Toda, Theory of Nonlinear Lattices, $2^{nd}$ enl. edition, Springer Series in Solid-State Sciences, 20, Springer, New York, 1994.

show all references

References:
[1]

V. I. Arnol'd, V. V. Kozlov and A. I. Neishtadt, Dynamical Systems III (Mathematical Aspects of Classical and Celestial Mechanics), Third edition, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.

[2]

R. F. Bikbaev and S. B. Kuksin, On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. Krichever, Lett. Math. Phys., 28 (1993), 115-122. doi: 10.1007/BF00750304.

[3]

E. Date and S. Tanaka, Analogue of inverse scattering theory for the discrete Hill's equation and exact solutions for the periodic Toda lattice, Progr. Theor. Phys., 55 (1976), 457-465. doi: 10.1143/PTP.55.457.

[4]

H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. Sect. B, 9 (1974), 1924-1925. doi: 10.1103/PhysRevB.9.1924.

[5]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems, in Collected Papers of Enrico Fermi, University of Chicago Press, Chicago, 2 (1965), 978-988; Theory, Methods and Applications, 2nd edition, Marcel Dekker, New York, 2000.

[6]

B. Grébert and T. Kappeler, Symmetries of the nonlinear Schrödinger equation, Bull. Soc. math. France, 130 (2002), 603-618.

[7]

A. Henrici and T. Kappeler, Global action-angle variables for the periodic Toda lattice, Int. Math. Res. Not., (2008), Art. ID rnn031, 52 pp. doi: 10.1093/imrn/rnn031.

[8]

A. Henrici and T. Kappeler, Global Birkhoff coordinates for the periodic Toda lattice, Nonlinearity, 21 (2008), 2731-2758. doi: 10.1088/0951-7715/21/12/001.

[9]

A. Henrici and T. Kappeler, Birkhoff normal form for the periodic Toda lattice, in Integrable Systems and Random Matrices, Contemp. Math., 458, American Mathematical Society, 2008, Providence, RI, 11-19.

[10]

A. Henrici and T. Kappeler, Results on normal forms for FPU chains, Comm. Math. Phys., 278 (2008), 145-177. doi: 10.1007/s00220-007-0387-z.

[11]

A. Henrici and T. Kappeler, Resonant normal form for even periodic FPU chains, J. Eur. Math. Soc., 11 (2009), 1025-1056. doi: 10.4171/JEMS/174.

[12]

A. Henrici and T. Kappeler, Nekhoroshev theorem for the periodic Toda lattice, Chaos, 19 (2009), 033120, 13 pp. doi: 10.1063/1.3196783.

[13]

T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation, Comm. Math. Phys., 285 (2009), 1087-1107. doi: 10.1007/s00220-008-0543-0.

[14]

T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[15]

T. Kappeler and P. Topalov, Global Well-Posedness of KdV in $H^{-1}(\mathbbT,\mathbbR)$, Duke Math. J., 135 (2006), 327-360. doi: 10.1215/S0012-7094-06-13524-X.

[16]

P. Lochak, Hamiltonian perturbation theory: Periodic orbits, resonances and intermittency, Nonlinearity, 6 (1993), 885-904. doi: 10.1088/0951-7715/6/6/003.

[17]

P. Lochak and A. Neishtadt, Estimates of stability time for nearly integrable systems with a quasi-convex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.

[18]

P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81. doi: 10.1007/BF01418827.

[19]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I, Uspekhi Mat. Nauk, 32 (1977), 5-66; Russian Math. Surveys, 32 (1977), 1-65.

[20]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems II, Trudy Sem. Petrovsk., 5 (1979), 5-50.

[21]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.

[22]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.

[23]

J. Pöschel, On Nekhoroshev's estimate at an elliptic equilibrium, Int. Math. Res. Not., 4 (1999), 203-215. doi: 10.1155/S1073792899000100.

[24]

B. Rink, Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice, Physica D, 175 (2003), 31-42. doi: 10.1016/S0167-2789(02)00694-2.

[25]

B. Rink, Proof of Nishida's conjecture on anharmonic lattices, Comm. Math. Phys., 261 (2006), 613-627. doi: 10.1007/s00220-005-1451-1.

[26]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys and Monographs, 72, Amer. Math. Soc., Providence, 2000.

[27]

M. Toda, Theory of Nonlinear Lattices, $2^{nd}$ enl. edition, Springer Series in Solid-State Sciences, 20, Springer, New York, 1994.

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