论文标题
关于立方衍生物NLS和Quintic NLS的周期性波的稳定性
On the stability of periodic waves for the cubic derivative NLS and the quintic NLS
论文作者
论文摘要
我们研究周期性的立方衍生物非线性schrödinger方程（DNLS）和（聚焦）五五次非线性schrödinger方程（NLS）。这些都是$ l^2 $关键分散模型，当在行$ {\ mathbb r} $上摆放时，它们表现出阈值类型行为。 我们以封闭形式描述了（三个参数）的非变化钟形解决方案。本文的主要目的是研究其相对于共同扰动的稳定性。我们分析了这些波是否在立方DNL框架中的稳定性。根据标量数量的迹象，我们提供了稳定性的标准。证明依赖于不稳定指数计数，这反过来依赖于对自相关矩阵山算子的详细光谱分析。我们在参数空间中展示了一个区域，该区域会产生光谱稳定的波。 我们还提供了对Quintic NLS的所有钟形行进波的稳定性的明确描述，事实证明，这是DNL展出的两个参数亚科。我们对它们的稳定性进行了完整的描述 - 事实证明，有些在频谱上是稳定的，而其他方面则在频谱上是不稳定的。
We study the periodic cubic derivative non-linear Schrödinger equation (dNLS) and the (focussing) quintic non-linear 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 a 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.
