Abstract
We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Bäcklund transformation and use the result to analyze the linearized FPU equation.
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References
Bambusi D., Ponno A.: On metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006)
Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences; American Mathematical Society, 2003
Eilbeck J.C., Flesch R.: Calculation of families of solitary waves on discrete lattices. Phys. Lett. A 149, 200–202 (1990)
Fermi, E., Pasta, J., Ulam, S.: Studies of Nonlinear Problems. I. Los Alamos Scientific Laboratory Report, LA-1940, 1955
Friesecke G., Pego R.L.: Solitary waves on FPU lattices, I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12, 1601–1627 (1999)
Friesecke G., Pego R.L.: Solitary waves on FPU 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)
Friesecke G., Wattis J.: Existence theorem for solitary waves on lattices. Commun. Math. Phys. 161, 391–418 (1994)
Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M.: Korteweg-deVries equation and generalization, VI. Methods for exact solution. Commun. Pure Appl. Math. 27, 97–133 (1974)
Haragus-Courcelle, M., Sattinger, D,H.: Inversion of the linearized Korteweg-deVries equation at the multi-soliton solutions. Z. Angew. Math. Phys. 49, 436–469 (1998)
Henry D.(1981) Geometric Theory of Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin
Hoffman A., Wayne C.E.: Counter-propagating two-soliton solutions in the Fermi-Pasta-Ulam lattice. Nonlinearity 21, 2911–2947 (2008)
Kapitula T.: On the stability of N-solitons in integrable systems. Nonlinearity 20, 879–907 (2007)
Kato T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in applied mathematics. Adv. Math. Suppl. Stud. 8, 93–128 (1983)
Maddocks J.H., Sachs R.L.: On the stability of KdV multi-solitons. Commun. Pure Appl. Math. 46, 867–901 (1993)
Martel Y., Merle F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rational Mech. Anal. 157, 219–254 (2001)
Martel Y., Merle F., Tsai T.P.: Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)
Martel Y., Merle F., Tsai T.P.: Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133, 405–466 (2006)
Mizumachi T.: Asymptotic stability of lattice solitons. Commun. Math. Phys. 288, 125–144 (2009)
Mizumachi T.: Weak interaction between solitary waves of the generalized KdV equations. SIAM J. Math. Anal. 35, 1042–1080 (2003)
Mizumachi T., Pego R.L.: Asymptotic stability of Toda lattice solitons. Nonlinearity 21, 2099–2111 (2008)
Mizumachi, T., Pego, R.L., Quintero, J.R.: Asymptotic stability of solitary waves in the Benney-Luke model of water waves. http://arxiv.org/pdf/1202.0450v1.pdf
Pego R.L., Weinstein M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)
Perelman G.: Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations. Spectral theory, microlocal analysis, singular manifolds. Math. Top. 14, 78–137 (1997)
Perelman G.: Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 29, 1051–1095 (2004)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness. Academic Press, New York, 1975
Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. Arxiv preprint math.AP/0309114 (2003)
Toda, M.: Theory of Nonlinear Lattices, 2nd edn. Springer Series in Solid-State Sciences, vol. 20. Springer, Berlin, 1989
Wahlquist H.D., Estabrook F.B.: Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16, 1–7 (1975)
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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Mizumachi, T. Asymptotic Stability of N-Solitary Waves of the FPU Lattices. Arch Rational Mech Anal 207, 393–457 (2013). https://doi.org/10.1007/s00205-012-0564-x
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DOI: https://doi.org/10.1007/s00205-012-0564-x