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

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 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))1/2 is cs2/24 then as 0 the renormalized displacement profile (1/2)rc(/) 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.