We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda > 0$, there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

中文翻译：

---

分数 KdV 和 NLS 方程的周期波稳定性

我们考虑聚焦分数周期 Korteweg-deVries (fKdV) 和分数周期非线性薛定谔方程 (fNLS) 方程，其中大号2亚临界色散。特别是，这涵盖了周期性 KdV 和 Benjamin-Ono 模型的情况。我们为 KdV（NLS 的驻波）构造了两个参数族的钟形行波，它们是哈密顿量的约束极小值。我们特别表明，对于每个$\lambda > 0$, fKdV 和 fNLS 有一个行波解$\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $，这是非退化的。我们还证明了波是光谱稳定和轨道稳定的，前提是柯西问题是局部适定的Hα/2[ -吨,吨] 和自然的技术条件。这是严格执行的，没有任何先验关于波浪平滑度或拉格朗日乘数的假设。