On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS
On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS
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DOI：
10.1007/s00332-021-09712-6
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发表时间：
2020-06
影响因子：
3
通讯作者：
S. Hakkaev;M. Stanislavova;A. Stefanov
中科院分区：
文献类型：
--
作者：
S. Hakkaev;M. Stanislavova;A. Stefanov
研究了周期三次导数的非线性薛定谔方程(DNLS)和聚焦的五次非线性薛定谔方程(NLS)。这两个都是关键的分散模型，当摆好姿势时，它们会表现出临界型行为。我们用闭合的形式描述了周期问题的(三参数)非零钟形解族。本文的主要目的是研究它们在共周期扰动下的稳定性。我们在三次DNLS的框架下分析了这些波的稳定性。我们根据标量的符号给出了稳定性的判据。证明依赖于不稳定指数计数，而不稳定指数计数又关键地依赖于对自伴矩阵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，它产生了频谱稳定的波。我们还给出了五次NLS的所有钟形行波的稳定性的显式描述，证明了它是DNLS的一个双参数子族。我们给出了它们的稳定性的完整描述--结果表明，对于共周期扰动，一些是谱稳定的，而另一些是谱不稳定的。
We study the periodic cubic derivative nonlinear Schrödinger equation (DNLS) and the (focussing) quintic nonlinear Schrödinger equation (NLS). These are bothcritical dispersive models, which exhibit threshold-type behavior, when posed on the line. 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.