Kernel nondegeneracy conjecture for periodic fractional Korteweg–de Vries waves

Let $\psi_0 \in H^{\alpha}_{\rm per}(\mathbb{T})$ be the solution to the boundary-value problem with $c=c_0$ obtained from the main existence theorem, and let $X_0$ and $\mathcal{L}$ denote the constrained subspace and linearized operator defined for this periodic wave. Kernel nondegeneracy conjecture. For every $c_0\in(-1,\infty)$ and every $\alpha\in\left(\frac{1}{3},2\right]$,

$$\operatorname{Ker}(\mathcal{L}|_{X_0})=\operatorname{span}(\partial_x\psi_0).$$

This conjecture asserts that translation is the only element of the constrained kernel for every periodic wave in the stated parameter range. The paper formulates it based on numerical evidence; establishing the claimed nondegeneracy would justify the kernel assumption used in the stability theory.