We study the periodic cubic derivative nonlinear Schrödinger equation (DNLS) and the (focussing) quintic nonlinear Schrödinger equation (NLS). These are both \(L^2\) critical dispersive models, which exhibit threshold-type behavior, when posed on the line \({{\mathbb {R}}}\). We describe the (three-parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two-parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability—as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.

我们研究了周期性三次导数非线性薛定equation方程（DNLS）和（聚焦）五次非线性薛定ding方程（NLS）。这些都是\（L ^ 2 \）关键色散模型，当置于线\（{{\ mathbb {R}}} \）上时，它们表现出阈值类型的行为。。我们以封闭形式描述周期问题的无消失钟形解决方案的（三参数）族。本文的主要目的是研究它们在同周期扰动方面的稳定性。我们在立方DNLS的框架中分析了这些波的稳定性。我们根据标量的符号提供稳定性标准。该证明依赖于不稳定指数计数，而不稳定指数反过来又严重依赖于对自伴矩阵Hill算子的详细频谱分析。我们在参数空间中展示了一个区域，该区域会产生频谱稳定的波。我们还提供了有关五次NLS的所有钟形行波稳定性的明确描述，结果证明这是DNLS所显示的一个二参数子族。