Periodic and homoclinic travelling waves in infinite lattices

Consider an infinite chain of particles subjected to a potential ff, where nearest neighbours are connected by nonlinear oscillators. The nonlinear coupling between particles is given by a potential VV. The dynamics of the system is described by the infinite system of second order differential equations q̈j+f′(qj)=V′(qj+1−qj)−V′(qj−qj−1),j∈Z. We investigate the existence of travelling wave solutions. Two kinds of such solutions are studied: periodic and homoclinic ones. On one hand, we prove under some growth conditions on ff and VV, the existence of non-constant periodic solutions of any given period T>0T>0, and speed c>c0c>c0, where the constant c0c0 depends on f″(0)f″(0) and V″(0)V″(0). On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed c>c0c>c0, emanating from the origin. Moreover, we prove that these homoclinics decay exponentially at infinity. Each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity.