Wavepackets and Nearly Solitary Waves in d-dimensional Periodic Media

In nonlinear periodic media of arbitrary dimension $d$ we consider the small amplitude asymptotics of wavepackets. The wavepackets have $N \in \mathbb{N}$ carrier Bloch waves of equal frequency. We use the cubic Gross-Pitaevskii equation (GP) as a prototype of the governing equation. We discuss two classical asymptotic scalings, one (for $N=1$) leading to the nonlinear Schroedinger equation and one (for $N>1$) leading to first order coupled mode equations (CMEs) as effective amplitude equations. Both of these models can support solitary waves - thus predicting nearly solitary waves of the GP. In particular, the CMEs for $d=1$ for the case of the coupling of two counter-propagating Bloch waves support a family of solitary waves parametrized by the velocity $v\in (-1,1)$. Can this be generalized to $d$ dimensions such that in the CMEs a solitary wave family parametrized by $\vec{v}\in (-1,1)^d$ exists? Solitary waves are typically found in spectral gaps. For $d\geq 2$ at least four ($N=4$) carrier waves are needed to produce CMEs with a spectral gap that supports solitary waves. However, only standing solitary waves have been found so far. We also provide a validity result of the $d-$dimensional NLS-asymptotics as well as the CME-asymptotics over asymptotically large time intervals.