Multiscale correction to solitary wave solutions on FPU lattices

Abstract

This paper looks at the behaviour of solitary waves in one-dimensional non-integrable lattices with the Hamiltonian H=∑j ((1/2)pj2 + V(qj+1-qj)), with generic nearest-neighbour potential V. It is known that solitary waves exist and that in the long-wave regime the profile is given to lowest order by a (re-scaled) KdV soliton. Here we determine by means of a rigorous expansion the leading order correction due to discreteness. This correction, and its governing equation, give an insight into how solitary waves in such FPU lattices differ from their integrable counterparts.