Periodic and homoclinic travelling waves in infinite lattices

Percy Makita

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Submission date: 20. Jul. 2009 (revised version: August 2009)
Pages: 24
published as:
Makita, P. D.: Periodic and homoclinic motions in infinite lattices
   Dissertation, Universität Leipzig, 2010
published as: Periodic and homoclinic travelling waves in infinite lattices.
In: Nonlinear analysis / A, 74 (2011) 6, p. 2071-2086 
DOI number (of the published article): 10.1016/j.na.2010.11.011
Bibtex
MSC-Numbers: 37K60, 34C25, 34C37
Keywords and phrases: Infinite dimensional Hamiltonian systems, Travelling waves, periodic and homoclinic motions
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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

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 , and any given speed . On the other hand, under very similar conditions, we establish the existence of non-trivial homoclinic solutions, of any given speed , emanating from the origin. Theses homoclinics are obtained as limits of periodic solutions by letting the period go to infinity.