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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
Jul 20, 2009
Published:
Aug 3, 2009
MSC Codes:
37K60, 34C25, 34C37
Keywords:
Infinite dimensional Hamiltonian systems, Travelling waves, periodic and homoclinic motions

Related publications

inJournal
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