Differential and Integral Equations

Existence of supersonic traveling waves for the Frenkel-Kontorova model

S. Issa, M. Jazar, and R. Monneau

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we study the standard one-dimensional (non-overdamped) Frenkel--Kontorova (FK) model describing the motion of atoms in a lattice. For this model we show that for any supersonic velocity $c>1$, there exist bounded traveling waves moving with velocity $c$. The profile of these traveling waves is a phase transition between limit states $k_-$ in $-\infty$ and $k_+$ in $+\infty$. Those limit states are some integers which reflect the assumed $1$-periodicity of the periodic potential inside the FK model. For every $c>1$, we show that we can always find $k_-$ and $k_+$ such that $k_+-k_-$ is an odd integer. Furthermore, for $c\ge \sqrt{\frac{25}{24}}$, we show that we can take $k_+-k_-=1$. These traveling waves are limits of minimizers of a certain energy functional defined on a bounded interval, when the length of the interval goes to infinity. Our method of proof uses a concentration-compactness-type argument which is based on a cleaning lemma for minimizers of this functional.

Article information

Source
Differential Integral Equations, Volume 26, Number 3/4 (2013), 321-353.

Dates
First available in Project Euclid: 5 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1360092827

Mathematical Reviews number (MathSciNet)
MR3059167

Zentralblatt MATH identifier
1289.34166

Subjects
Primary: 35A15: Variational methods 35C07: Traveling wave solutions 37K60: Lattice dynamics [See also 37L60]

Citation

Issa, S.; Jazar, M.; Monneau, R. Existence of supersonic traveling waves for the Frenkel-Kontorova model. Differential Integral Equations 26 (2013), no. 3/4, 321--353. https://projecteuclid.org/euclid.die/1360092827


Export citation