## Differential and Integral Equations

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

### Existence of supersonic traveling waves for the Frenkel-Kontorova model

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

#### 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