Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit

G. Friesecke, R. L. Pego

Publikation: Beitrag in FachzeitschriftArtikelBegutachtung

179 Zitate (Scopus)

Abstract

This paper is the first in a series to address questions of qualitative behaviour, stability and rigorous passage to a continuum limit for solitary waves in one-dimensional non-integrable lattices with the Hamiltonian H = ∑j∈ℤ (1/2 p2j + V (qj+1 - qj)), with a generic nearest-neighbour potential V. Here we establish that for speeds close to sonic, unique single-pulse waves exist and the profiles are governed by a continuum limit valid on all length scales, not just the scales suggested by formal asymptotic analysis. More precisely, if the deviation of the speed c from the speed of sound cs = √V″(0) is csε2/24 then as ε → 0 the renormalized displacement profile (1/ε2)rc(·/ε) of the unique single-pulse wave with speed c, qj+1(t) - qj(t) = rc(j - ct), is shown to converge uniformly to the soliton solution of a KdV equation containing derivatives of the potential as coefficients, -rx + rxxx + 12(V‴(0)/V″(0))r rx = 0. Proofs involve (a) a new and natural framework for passing to a continuum limit in which the above KdV travelling-wave equation emerges as a fixed point of a renormalization process, (b) careful singular perturbation analysis of lattice Fourier multipliers and (c) a new Harnack inequality for nonlinear differential-difference equations.

OriginalspracheEnglisch
Seiten (von - bis)1601-1627
Seitenumfang27
FachzeitschriftNonlinearity
Jahrgang12
Ausgabenummer6
DOIs
PublikationsstatusVeröffentlicht - Nov. 1999
Extern publiziertJa

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