论文标题
弱非线性拓扑系统中的孤子：线性化，e象同谋和K理论
Solitons in Weakly Non-linear Topological Systems: Linearization, Equivariant Cohomology and K-theory
论文作者
论文摘要
缺乏有关具有周期性潜力的非线性$ d $维系统的拓扑不变的知识。我们通过对孤子溶液周围线性化的NLS/GP方程进行分类来研究这些系统。线性化（模式）绝热进化下的稳定条件可以通过拓扑解释，我们可以使用Equivariant $ \ Mathit {k} $ - 理论和共同体学进行分类。在带有晶体学点组$ p $的晶格上，稳定的模式，$ p $ - 对称孤子由$ \ bar {\ mathit {k}} _ {p} _ {p}^{0，τ}（\ mathbb {t}^d}^d）\ oplus h^2（bp;同样，对于$ p $ -smmetric Gap soliton的振荡稳定性，我们有$ \ bar {\ mathit {k}} _ {p}^{0，τ}（\ mathbb {t}^d）如果我们包括边界，则可以用$ \ m artiT {t}^{p}^{0，τ}（\ Mathbb {t}^d）$替换$ \ bar {\ mathit {K}} _ {p}^{0，τ}最后，我们提到了如何使用这些方法，以及Soliton Solutions $ M_ {D}（e_ {gap}）$和$ m_ {o}（e_ {gap}）$的空间，为系统提供全局不变性。
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 $\bar{\mathit{K}}_{P}^{0,τ}(\mathbb{T}^d)\oplus H^2(BP;\mathbb{Z})$. Similarly, for $P$-symmetric gap solitons that are oscillatory stable, we have $\bar{\mathit{K}}_{P}^{0,τ}(\mathbb{T}^d)\oplus \tilde{R}(P)$ instead. If we include a boundary, we can replace $\bar{\mathit{K}}_{P}^{0,τ}(\mathbb{T}^d)$ with $\mathit{K}^{-1,τ}_{P}(\mathbb{T}^{d-1})$. 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.
