Solitons in Weakly Non-linear Topological Systems: Linearization, Equivariant Cohomology and K-theory

Abstract

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 $K$-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 ${\bar{K}}_P^{0,\ddot{I}„}({\mathbb{T}}^d)\oplus H^2(BP;\mathbb{Z})$. Similarly, for $P$-symmetric gap solitons that are oscillatory stable, we have ${\bar{K}}_P^{0,\ddot{I}„}({\mathbb{T}}^d)\oplus \tilde{R}(P)$ instead. If we include a boundary, we can replace ${\bar{K}}_P^{0,\ddot{I}„}({\mathbb{T}}^d)$ with $K_P^{-1,\ddot{I}„}({\mathbb{T}}^{d-1})$. 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.

6 pages, 1 figure