  There is a lack of knowledge about the topological invariants of non-linear
$d$-dimensional systems with a periodic potential. We study these systems
through a classification of the linearized NLS/GP equation around their soliton
solutions. Stability conditions under linearized (mode) adiabatic evolution can
be interpreted topologically and we can use equivariant $\mathit{K}$-theory and
cohomology for their classification. On a lattice with crystallographic point
group $P$, modes around stable, $P$-symmetric solitons are coarsely classified
by the groups $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus
H^2(BP;\mathbb{Z})$. Similarly, for $P$-symmetric gap solitons that are
oscillatory stable, we have $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus
\tilde{R}(P)$ instead. If we include a boundary, we can replace
$\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)$ with
$\mathit{K}^{-1,\tau}_{P}(\mathbb{T}^{d-1})$. Finally, we mention how to use
these, and the spaces of soliton solutions $M_{D}(E_{gap})$ and
$M_{O}(E_{gap})$ to provide global invariants for the system.
