Abstract
We present an analytical study of the Fermi–Pasta–Ulam (FPU) α–model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with one low frequency Fourier mode excited. We show that, correspondingly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy.
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Bambusi, D.: Nekhoroshev theorem for small amplitude solution sin nonlinear Schrödinger equation. Math. Z. 130, 345–387 (1999)
Bambusi, D.: An averaging theorem for quasilinear Hamiltonian PDEs. Ann. Henri Poincaré 4, 685–712 (2003)
Bambusi, D.: Galerkin averaging method and Poincaré normal form for some quasilinear PDEs. http://www.ma.utexas.edu/mp_arc/c/05/05-28.pdf, 2005
Bambusi, D., Carati, A., Ponno, A.: The nonlinear Schrødinger equation as a resonant normal form. DCDS-B 2, 109–128 (2002)
Berchialla, L., Galgani, L., Giorgilli, A.: Localization of energy in FPU chains. DCDS-A 11, 855–866 (2005)
Berchialla, L., Giorgilli, A., Paleari, S.: Exponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A 321, 167–172 (2004)
Biello, J.A., Kramer, P.R., LvovD, Y.V.: Stages of energy transfer in the FPU model. Dynamical systems and differential equations (Wilmington NC 2002). DCDS Suppl., 113–122 (2003)
Bambusi, D., Nekhoroshev, N.N.: A property of exponential stability in the nonlinear wave equation close to main linear mode. Physica D 122, 73–104 (1998)
Carati, A., Galgani, L.: On the specific heat of FPU systems and their glassy behavior. J. Stat. Phys. 94, 859–869 (1999)
Carati, A., Galgani, L.: Planck's formula and glassy behaviour in classical nonequilibrium statistical mechanics. Physica A 280, 105–114 (2001)
Carati, A., Galgani, L., Giorgilli, A.: The Fermi–Pasta–Ulam problem as a challenge for the foundations of physics. Chaos, to appear, 2005
Craig, W.: Birkhoff normal form for water waves. Mathematical problems in the theory of water waves, V. 200, Providence, EI: AMS, 1996
Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)
Craig, W., Worfolk, P.A.: An integrable normal form for water waves in infinite depth. Physica D 84, 513–531 (1995)
Dyachenko, A.I., Zakharov, V.E.: Is free-surface hydrodynamics an integrable system?. Phys. Lett. A 190, 144–148 (1994)
Fink, A.: Almost periodic differential equations. Berlin: Springer-Verlag, 1974
Fucito, F., Marchesoni, F., Marinari, E., Parisi, G., Peliti, L., Ruffo, S., Vulpiani, A.: Approach to equilibrium in a chain of nonlinear oscillators. J. de Physique 43, 707–713 (1982)
Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. I. Qualitative properties renormalization and continuum limit. Nonlinearity 12, 1601–1627 (1999)
Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. II. Linear implies nonlinear stability. Nonlinearity 15, 1343–1359 (2002)
Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. III. Howland-type Floquet theory. Nonlinearity 17, 207–227 (2004)
Friesecke, G., Pego, R.L.: Solitary waves on Fermi-Pasta-Ulam lattices. IV. Proof of stability at low energy. Nonlinearity 17, 229–251 (2004)
Fermi, E., Pasta, J.R., Ulam, S.M.: Studies of nonlinear problems. In Collected works of E. Fermi Vol.2. Chicago: Chicago University Press, 1965
Galgani, L., Scotti, A.: Planck-like distribution in classical nonlinear mechanics. Phys. Rev. Lett. 28, 1173–1176 (1972)
Izrailev, F.M., Chirikov, B.V.: Statistical properties of a nonlinear string. Sov. Phys. Dokl. 11, 30–32 (1966)
Kappeler, T. Pöschel, J.: KAM & KdV. Berlin-Heidelberg-Newyork: Springer, 2003
Livi, R., Pettini, M., Ruffo, S., Vulpiani, A.: Further results on the equipartition threshold in large nonlinear Hamiltonian systems. Phys. Rev. A 31, 2741–2742 (1985)
Marchenko, V.: Sturm-Liouville operators and applications. Basel: Birkhäuser, 1986
Ponno, A., Bambusi, D.: Energy cascade in Fermi–Pasta–Ulam model. In: G. Gaeta et al. (eds.) Symmetry and Perturbation Theory 2004, RiverEdge, NJ: World Scientific, 2005 pp. 263–270
Ponno, A., Bambusi, D.: KdV equation and energy sharing in FPU. Chaos 15, 015107 (2005)
Paleari, S., Bambusi, D., Cacciatori, S.: Normal form and exponential stability for some nonlinear string equations. ZAMP 52, 1033–1052 (2001)
Pettini, M., Landolfi, M.: Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics. Phys. Rev. A 41, 768–783 (1990)
Ponno, A.: Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit. Europhys. Lett. 64, 606–612 (2003)
Ponno, A.: The Fermi–Pasta–Ulam problem in the thermodynamic limit. In: P. Collet et al. (ed.) Proceedings of the Cargése Summer School 2003 on Chaotic Dynamics and Transport in Classical and Quantum Systems, Dordrecht: Kluwer Academic Publishers, 2005, pp. 431–440
Pöschel, J.: Hill's potentials in weighted Sobolev spaces and their spectral gaps. Preprint (2004)
Pierce, R.D., Wayne, C.E.: On the validity of mean-field amplitude equations for counterpropagating wavetrains Nonlinearity 8, 769–780 (1995)
Rink, B.: Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice. Physica D 175, 31–42 (2001)
Rink, B.: Symmetry and resonance in periodic FPU chains. Commun. Math. Phys. 218, 665–685 (2001)
Shepelyansky, D.L.: Low-Energy chaos in the Fermi–Pasta–Ulam problem. Nonlinearity 10, 1331–1338 (1997)
Schneider, G., Wayne, C.E.: Counter-propagating waves on fluid surfaces and the continuun limit of the Fermi Pasta Ulam model. In: Proceedings of the International Conference on Differential Equations Berlin 1999, River Edge NJ : World Scientific, 2000
Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)
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Bambusi, D., Ponno, A. On Metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006). https://doi.org/10.1007/s00220-005-1488-1
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DOI: https://doi.org/10.1007/s00220-005-1488-1