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

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 = Sigma(j is an element of Z) (1/2p(j)(2) + V(q(i+1) - q(j))), with a generic nearest-neighbour potential V. Here we establish that for speeds close to sonic, unique single-purse 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 c(s) = root V "(0) is c(s)epsilon(2)/24 then as epsilon --> 0 the renormalized displacement profile (1/epsilon(2))r(c)(./epsilon) Of the unique single-pulse wave with speed c, q(j+1)(t) - q(j)(t) = r(c)(j - ct), is shown to converge uniformly to the soliton solution of a KdV equation containing derivatives of the potential as coefficients, -r(x) + r(xxx) + 12(V'''(0)/V "(0))rr(x) = 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. AMS classification scheme numbers: 70F, 70H, 76B25, 35Q51, 35Q53, 82B28.