March/April 2013 Existence of supersonic traveling waves for the Frenkel-Kontorova model
S. Issa, M. Jazar, R. Monneau
Differential Integral Equations 26(3/4): 321-353 (March/April 2013). DOI: 10.57262/die/1360092827


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.


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S. Issa. M. Jazar. R. Monneau. "Existence of supersonic traveling waves for the Frenkel-Kontorova model." Differential Integral Equations 26 (3/4) 321 - 353, March/April 2013.


Published: March/April 2013
First available in Project Euclid: 5 February 2013

zbMATH: 1289.34166
MathSciNet: MR3059167
Digital Object Identifier: 10.57262/die/1360092827

Primary: 35A15 , 35C07 , 37K60

Rights: Copyright © 2013 Khayyam Publishing, Inc.


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Vol.26 • No. 3/4 • March/April 2013
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