Interaction of modulated pulses in scalar multidimensional nonlinear lattices

TL;DR

This paper derives and rigorously justifies macroscopic evolution equations for weakly amplitude-modulated pulses in multidimensional nonlinear lattices, analyzing their interactions and resonance conditions.

Contribution

It provides a formal multiscale derivation of pulse interaction equations in scalar nonlinear lattices, including resonance analysis and rigorous justification.

Findings

Explicit evolution equations for pulse amplitudes derived

Complete interaction list for up to three pulses presented

Resonance and non-resonance conditions identified

Abstract

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.