On the impact of FPU lattice discreteness upon solitary wave interactions

This work is concerned with the motions of an infinite one-dimensional lattice with nearest-neighbour interactions governed by a generic potential. The Hamiltonian of such a system may be written H = ∑ i = − ∞ ∞ 1 2 p i 2 + V ( q i + 1 − q i ) , in terms of the momenta p i and the displacements q i of the lattice sites. All sites are assumed to be of equal mass. Certain generic conditions are placed on the potential V. The particular results of this paper concern the way in which lattice discreteness impacts upon the solitary wave interaction process, which is known to be well approximated in the long-wave continuum limit by KdV soliton interaction. An evolution equation for discreteness effects on the lattice is proposed and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion-acoustic waves. It is proved, by standard techniques, to have a unique classical solution, whose profile is explicitly obtained by numerical simulation. The most important outcome is the observation that lattice discreteness imposes an effective repulsion between post-collision solitary waves, relative to their unperturbed “pure-KdV” positions.