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

TL;DR

This paper develops a topological classification framework for solitons in weakly non-linear systems using equivariant K-theory and cohomology, revealing new invariants related to stability and symmetry.

Contribution

It introduces a novel topological classification scheme for solitons in non-linear systems based on equivariant K-theory and cohomology, extending understanding of their invariants.

Findings

Classification of modes around stable solitons using K-theory and cohomology.

Identification of topological invariants for gap solitons with symmetry considerations.

Extension of classification to systems with boundaries and global invariants.

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 $\mathit{K}$-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.