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 & Continuous Dynamical Systems - A, 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, (2006). Google Scholar

[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. doi: 10.1007/BF00750304. Google Scholar

[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. doi: 10.1143/PTP.55.457. Google Scholar

[4]

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

[5]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems,, in Collected Papers of Enrico Fermi, 2 (1965), 978. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Henrici and T. Kappeler, Birkhoff normal form for the periodic Toda lattice,, in Integrable Systems and Random Matrices, (2008), 11. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[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], (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I,, Uspekhi Mat. Nauk, 32 (1977), 5. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices,, Math. Surveys and Monographs, 72 (2000). Google Scholar

[27]

M. Toda, Theory of Nonlinear Lattices,, $2^{nd}$ enl. edition, (1994). Google Scholar

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, (2006). Google Scholar

[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. doi: 10.1007/BF00750304. Google Scholar

[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. doi: 10.1143/PTP.55.457. Google Scholar

[4]

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

[5]

E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems,, in Collected Papers of Enrico Fermi, 2 (1965), 978. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Henrici and T. Kappeler, Birkhoff normal form for the periodic Toda lattice,, in Integrable Systems and Random Matrices, (2008), 11. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[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], (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I,, Uspekhi Mat. Nauk, 32 (1977), 5. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices,, Math. Surveys and Monographs, 72 (2000). Google Scholar

[27]

M. Toda, Theory of Nonlinear Lattices,, $2^{nd}$ enl. edition, (1994). Google Scholar

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