MiS Preprint Repository

Consider an infinite lattice of particles in one dimension subjected to a potential $f$ and such that each site interacts (only) with its nearest neighbours under an interaction potential $V$. The dynamics of the system is described by the infinite system of second order differential equations $${\ddot{q}}_j+f^{\prime }(q_j)=V^{\prime }(q_{j+1}-q_j)-V^{\prime }(q_j-q_{j-1}),j\in \mathbb{Z}.$$ We investigate the existence of travelling wave solutions. Two kinds of such solutions are studied: periodic and homoclinic ones. On the one hand, we prove under some growth conditions on $f$ and $V$, the existence of non-constant periodic solutions of any given period $\tau >0$, and any given speed $c>c_0$. On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed $c>c_0$, emanating from the origin. Theses homoclinics are obtained as limits of periodic solutions by letting the period go to infinity.