Existence theorem for solitary waves on lattices

Gero Friesecke, Jonathan A D Wattis

Research output: Contribution to journalArticle

Abstract

In this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian {Mathematical expression} where V(·) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda lattice V(f)=ab-1(e-bf+bf-1), the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic lattices V(f)= 1/2 f2 + 1/3 af3, V(f) = 1/2 f2 + 1/4 bf4, thereby confirming the numerical findings in [1] and shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam [2], and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but "generic" in this class of nonlinear lattices. The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamilton's principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions [3]). © 1994 Springer-Verlag.

Original languageEnglish
Pages (from-to)391-418
Number of pages28
JournalCommunications in Mathematical Physics
Volume161
Issue number2
DOIs
Publication statusPublished - Mar 1994

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