Interaction of modulated pulses in scalar multidimensional nonlinear lattices

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 nonlinear interaction potentials of arbitrary range 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 nonresonance 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 of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

Keywords:

• nonlinear discrete lattices
• interaction of modulated pulses
• multiscale ansatz
• derivation and justification of macroscopic dynamics

AMS Subject Classifications::

• 37K60
• 34E13
• 34C20
• 70F45
• 70K70
• 35L45
• 35L60

Acknowledgements

This work has been partially supported by the DFG Priority Program 1095 Analysis, Modeling and Simulation of Multiscale Problems under Mi 459/3–3. I thank Alexander Mielke, who suggested this idea to me, and Michael Herrmann for the many fruitful discussions.