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, 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 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.

Transport and generation of macroscopically modulated waves in diatomic chains

“…The present work constitutes a generalization of previous work of the author, see [3], to a case of vector-valued displacement in nonlinear lattices. As the technically most simple but yet generic case we consider a nonlinear diatomic chain.…”

“…and using the corollary of Sobolev's embedding theorem (cf., e.g.,[3, Lemma 3.1])ϕ ε(• + ξ) ℓ 2 ≤ cε −1/2 ϕ H 1 (R;C) , ξ : Z → [−1, 1], ε ∈ (0, ε 0 ], and A, B ∈ C([0, τ 0 ]; H 1 (R; C)) ⇒ AB ∈ C([0, τ 0 ]; H 1 (R; C)), we obtain that the above estimates are satisfied, provided C([0, τ 0 ]; H 1 (R; C)) for |(p, q)| ≤ 2, (p, q) = (3, 0), (2, 1).…”

Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

“…Concerning nonlinear wave interactions, there exist several results (mostly three-wave mixing) in the context of strictly hyperbolic systems, see, e.g., [18,19,22,25], and in the case of microscopically discrete dynamical systems, cf. [11,12] and the references given therein. For spatially periodic systems we again refer to [27] and to [1], where a similar system of coupled-mode equations is derived from Maxwell's equations in photonic crystals.…”

Interaction of modulated gravity water waves of finite depth

“…Of course, more generally one could consider also an arbitrary number of N ∈ N, N 3, modulated plane waves, as was done for other physical settings, e.g. in [23,22]. In order to keep the presentation more simple and explicit, we chose, however, to consider here only three pulses.…”