Existence of supersonic traveling waves for the Frenkel-Kontorova model. (English) Zbl 1289.34166

The paper deals with the following infinite system of differential equations \[ {\ddot q}_n=q_{n+1}+q_{n-1}-2q_n+W'(q_n). \] It is supposed that the nonlinear potential is \(C^2\)- and \(1\)-periodic. Furthermore, \(W(a)>0=W(0)=W'(0)\) for every \(a\in \mathbb{R}\setminus \mathbb{Z}\) and \(W''(0)>0\). A traveling wave is a solution of the form \(q_n(t)=u(n-ct)\), where \(c\) is the speed of the wave. The profile function \(u\) satisfies certain advance-delay differential equation. The authors look for solutions satisfying the following conditions at infinity \[ u(\pm \infty )\in \mathbb{Z},\quad u(+\infty )-u(-\infty )\in 1+2\mathbb{Z}. \] The main result shows that for any \(c>1\) there exists such a wave. These are supersonic waves. Furthermore, if \(c\geq \sqrt {25/24}\), then there exists a traveling wave with \(u(+\infty )-u(-\infty )=1\).


34K10 Boundary value problems for functional-differential equations
34A33 Ordinary lattice differential equations
35C07 Traveling wave solutions
35Q53 KdV equations (Korteweg-de Vries equations)