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

Michael Herrmann, Karsten Matthies

Research output: Contribution to journalArticle

Abstract

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a longstanding 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 wave 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.
LanguageEnglish
Pages3425–3486
Number of pages62
JournalTransactions of the American Mathematical Society
Volume372
Issue number5
Early online dateJan 2019
DOIs
StatusE-pub ahead of print - Jan 2019

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Solitary Waves
Solitons
High Energy
Unstable
Eigenvalues and eigenfunctions
Eigenfunctions
Disprove
Essential Spectrum
Local Existence
Nonlinear Stability
Weighted Spaces
Korteweg-de Vries Equation
Eigenvalue Problem
Open Problems
Uniqueness
Complement
Linearly
Singularity
Wavelength
Eigenvalue

Keywords

  • math.DS
  • 37K60, 37K40, 70H14, 74H10

Cite this

Stability of high-energy solitary waves in Fermi-Pasta-Ulam-Tsingou chains. / Herrmann, Michael; Matthies, Karsten.

In: Transactions of the American Mathematical Society, Vol. 372, No. 5, 01.09.2019, p. 3425–3486.

Research output: Contribution to journalArticle

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AB - The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a longstanding 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 wave 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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