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Transactions of the American Mathematical Society

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Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains

Authors: Michael Herrmann and Karsten Matthies
Journal: Trans. Amer. Math. Soc. 372 (2019), 3425-3486
MSC (2010): Primary 37K60; Secondary 37K40, 70H14, 74H10
Published electronically: June 6, 2019
MathSciNet review: 3988616
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The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a long-standing open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. In this paper we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result and the extension by Mizumachi.

Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy waves to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is no unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones.

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Additional Information

Michael Herrmann
Affiliation: Technische Universität Braunschweig, Institute of Computational Mathematics, 38092 Braunschweig, Germany
MR Author ID: 933936

Karsten Matthies
Affiliation: Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
MR Author ID: 653313

Keywords: Hamiltonian lattices waves, high-energy limit of FPU or FPUT chains, nonlinear orbital stability, asymptotic analysis
Received by editor(s): September 4, 2017
Received by editor(s) in revised form: November 26, 2018
Published electronically: June 6, 2019
Additional Notes: Both authors are grateful for the support by the Deutsche Forschungsgemeinschaft (DFG individual grant HE 6853/2-1) and the London Mathematical Society (LMS Scheme 4 Grant, Ref 41326)
The second author also thanks the University of Münster for the kind hospitality during his sabbatical stay
Article copyright: © Copyright 2019 American Mathematical Society