论文标题
关于本尼系统周期波的稳定性
On the stability of the periodic waves for the Benney system
论文作者
论文摘要
我们分析了本尼模型在共振水波相互作用中短波和长波的相互作用。我们特别感兴趣的是我们详细构建和研究的周期性流动波。主要的结果是，对于参数的所有自然值，相对于同一时期的扰动，周期性的脱水波在光谱上都是稳定的。对于另一组自然参数，我们在同一设置中构造了表现出不稳定性的旋转波。 我们的结果是在这种情况下的第一个不稳定结果。另一方面，本文建立的光谱稳定性大大改善了Angulo-Corcho-Hakkaev的工作，该工作通过依靠Grillakis-Shatah理论，在参数空间的一个子集中确立了Dnoidal波的稳定性。事实证明，我们的方法为参数的整个领域提供了明确的答案，依赖于不稳定索引理论。有趣的是，即使线性化运算符是明确的，我们的频谱分析也需要对矩阵Schrödinger运算符的微妙而详细的分析，这支持了一些有趣的功能。
We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly upon the work Angulo-Corcho-Hakkaev, which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis-Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schrödinger operators in the periodic context, which support some interesting features.
