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 = ∑_j∈ℤ (1/2 p²_j + V (q_j+1 - q_j)), 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 c_s = √V″(0) is c_sε²/24 then as ε → 0 the renormalized displacement profile (1/ε²)r_c(·/ε) 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))r r_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.