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MiS Preprint

Periodic and homoclinic travelling waves in infinite lattices

Percy Makita


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.

MSC Codes:
37K60, 34C25, 34C37
Infinite dimensional Hamiltonian systems, Travelling waves, periodic and homoclinic motions

Related publications

2011 Repository Open Access
Percy D. Makita

Periodic and homoclinic travelling waves in infinite lattices

In: Nonlinear analysis / A, 74 (2011) 6, pp. 2071-2086