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Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

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The Fermi-Pasta-Ulam Problem

Part of the book series: Lecture Notes in Physics ((LNP,volume 728))

Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi–Pasta–Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete Klein–Gordon equations.

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References

  1. E. Fermi, J. Pasta and S. Ulam, Technical Report LA-1940, Los Alamos National Laboratory (1955).

    Google Scholar 

  2. N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).

    Article  ADS  MATH  Google Scholar 

  3. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967).

    Article  MATH  ADS  Google Scholar 

  4. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Commun. Pure Appl. Math 27, 97 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  5. L.A. Kalyakin, Russian Math. Surveys 44(1), 3 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. G. Schneider and C.E. Wayne, in B. Fiedler, K. Gröger and J. Sprekels, eds, International Conference on Differential Equations Appl. 5(1), 69 (1998).

    Google Scholar 

  7. A. Ponno and D. Bambusi, Chaos 15, 015107 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  8. G. Friesecke and J. Wattis, Commun. Math. Phys. 161, 391 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. G. Friesecke and K. Matthies, Physica D 171, 211 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. D. Treschev, Disc. Cont. Dyn. Syst. A 11, 867 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Smets and M. Willem, J. Funct. Anal. 149, 266 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Pankov and K. Pflüger, Math. Meth. Appl. Sci. 23, 1223 (2000).

    Article  MATH  Google Scholar 

  13. G. Friesecke and R.L. Pego, Nonlinearity 12, 1601 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. G. Friesecke and R.L. Pego, Nonlinearity 15, 1343 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. G. Friesecke and R.L. Pego, Nonlinearity 17, III:207, IV:229 (2004).

    ADS  MathSciNet  Google Scholar 

  16. G. Iooss, Nonlinearity 13, 849 (2000).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. D.E. Pelinovsky and V.M. Rothos, Physica D 202, 16 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. G.Iooss and K.Kirchgässner, Com. Math. Phys. 211, 439 (2000).

    Google Scholar 

  19. Y. Sire and G. James, C.R. Acad. Sci. Paris, 338, Série I, 661 (2004).

    MathSciNet  Google Scholar 

  20. G. James and Y. Sire, Commun. Math. Phys. 257, 51 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Y. Sire, J. Dyn. Diff. Eqs. 17, 4 (2005).

    Article  MathSciNet  Google Scholar 

  22. J.P. Boyd, Nonlinearity 3, 177 (1990).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. J. Giannoulis and A. Mielke, Nonlinearity 17, 551 (2004).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Y. Sire and G. James, Physica D 204, 15 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. S. Aubry and T. Crétégny, Physica D 119, 34 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. A.V. Savin, Y. Zolotaryuk, and J.C. Eilbeck, Physica D 138, 267 (2000).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. G. Iooss and G. James. Chaos 15, 015113 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  28. G. James, J. Nonlinear Sci. 13(1), 27 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. G. James, C.R. Acad. Sci. Paris 332, Série I, 581 (2001).

    ADS  Google Scholar 

  30. G. Arioli and A. Szulkin, Ann. Sci. Math. Québec 22, 97 (1998).

    MATH  MathSciNet  Google Scholar 

  31. S. Aubry, G. Kopidakis and V. Kadelburg, Disc. Cont. Dyn. Syst. B 1, 271 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  32. G. James and P. Noble, Physica D 196, 124 (2004).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. P. Noble, Nonlinearity 17, 1 (2004).

    Article  MathSciNet  Google Scholar 

  34. B. Sànchez-Rey, G. James, J. Cuevas and J.F.R. Archilla, Phys. Rev. B 70, 014301 (2004).

    Article  ADS  Google Scholar 

  35. A. Mielke, Math. Meth. Appl. Sci. 10, 51 (1988).

    Google Scholar 

  36. A. Vanderbauwhede and G. Iooss, Dynamics Reported 1, New Series, C. Jones, U. Kirchgraber and H. Walther eds, Springer Verlag, 125 (1992).

    MathSciNet  Google Scholar 

  37. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, NY, 1983.

    MATH  Google Scholar 

  38. G. Iooss, Bifurcation of maps and applications, Math. Studies 36, Elsevier-North-Holland, Amsterdam, 1979.

    Google Scholar 

  39. J. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer Verlag, NY, 1976.

    MATH  Google Scholar 

  40. X. Cabré, E. Fontich and R. de la Llave, Indiana Univ. Math. J. 52, I:283, II:329 (2003).

    Google Scholar 

  41. X. Cabré, E. Fontich and R. de la Llave, J. Diff. Eqs., 218, 2 (2005).

    Article  Google Scholar 

  42. K. Kirchgässner, J. Diff. Eqs. 45, 113 (1982).

    Google Scholar 

  43. S. Flach and C.R. Willis, Phys. Rep. 295, 181 (1998).

    Article  MathSciNet  ADS  Google Scholar 

  44. T. Kato, Perturbation theory for linear operators, Springer Verlag (1966).

    Google Scholar 

  45. G. Iooss and M. Adelmeyer, Topics in bifurcation theory and applications, Adv. Series in Nonlinear Dyn. 3, World Sci. (1992).

    Google Scholar 

  46. A. Vanderbauwhede, Dynamics Reported 2, in U. Kirchgraber and H.O. Walther, eds, John Wiley and Sons Ltd and B.G. Teubner, 89 (1989).

    Google Scholar 

  47. A. Mielke, Math. Ann. 277, 121 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  48. D.K. Arrowsmith and C.M. Place, An introduction to dynamical systems, Cambridge University Press, 1990.

    Google Scholar 

  49. A. Alvarez, J.F.R. Archilla, J. Cuevas and F.R. Romero, New Journal of Physics 4, 72 (2002).

    Article  ADS  Google Scholar 

  50. A. Tsurui, Prog. Theor. Phys. 48(4), 1196 (1972).

    Article  ADS  Google Scholar 

  51. S. Flach: Physica D 91, 223 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  52. S.R. Bickham, S.A. Kiselev and A.J. Sievers, Phys. Rev. B 47, 14206 (1993).

    Article  ADS  Google Scholar 

  53. S. Flach and A. Gorbach, Chaos 15, 015112 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  54. G. Iooss and M-C Pérouéme, J. Diff. Eqs. 102, 62 (1993).

    Article  MATH  Google Scholar 

  55. E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems, Lecture Notes in Mathematics 1741, Springer-Verlag (2000).

    Google Scholar 

  56. M.J.D. Powell, numerical methods for nonlinear algebraic equations, Gordon and Breach (1970).

    Google Scholar 

  57. H. Feddersen, M. Remoissenet and M. Peyrard, eds, nonlinear coherent structures in physics and biology, Lecture Notes in Physics 393, 159, Springer-Verlag, 1991.

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire Mathématiques pour l’Industrie et la Physique UMR CNRS 5640, INSA de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France

    Guillaume James & Yannick Sire

Authors
  1. Guillaume James
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  2. Yannick Sire
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Editors and Affiliations

  1. Dipartimento di Matematica, Università Roma La Sapienza, Piazzale Aldo Moro, 2 00185 Roma, Italy

    Giovanni Gallavotti

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James, G., Sire, Y. (2007). Center Manifold Theory in the Context of Infinite One-Dimensional Lattices. In: Gallavotti, G. (eds) The Fermi-Pasta-Ulam Problem. Lecture Notes in Physics, vol 728. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72995-2_6

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