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MiS Preprint
36/2009
Periodic and homoclinic travelling waves in infinite lattices
Percy Makita
Abstract
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 $$ \ddot{q}_{j}+f'(q_{j})=V'(q_{j+1}-q_{j})-V'(q_{j}-q_{j-1}),\, j \in \Bbb Z. $$ 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.
