The Ostrovskyi (Ostrovskyi-Vakhnenko/short pulse) equations are ubiquitous models in mathematical physics. They describe water waves under the action of a Coriolis force as well as the amplitude of a "short" pulse in an optical fiber. In this paper, we rigorously construct ground traveling waves for these models as minimizers of the Hamiltonian functional for any fixed $L^2$ norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the non-triviality of the limits of the minimizing sequences. We show that (at least almost all of) the waves are strongly spectrally stable, along with other properties: smoothness with respect to parameters, weak non-degeneracy of the waves etc. In the case of a quadratic nonlinearities, it is known that these waves are unique, by work of Zhang-Liu. Thus, our results imply in particular, that all traveling waves for the classical Ostrovsky equation are spectrally stable.

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关于 Ostrovskyi 方程的基态及其稳定性

Ostrovskyi（Ostrovskyi-Vakhnenko/短脉冲）方程是数学物理学中无处不在的模型。他们描述了在科里奥利力作用下的水波以及光纤中“短”脉冲的幅度。在本文中，我们为这些模型严格构建了地面行波，作为任何固定 $L^2$ 范数的哈密顿函数的最小值。存在性论证通过补偿紧致性方法进行，但它需要令人惊讶的详细傅立叶分析论证来排除最小化序列极限的非平凡性。我们表明（至少几乎所有）波在光谱上是非常稳定的，以及其他特性：关于参数的平滑性、波的弱非简并性等。在二次非线性的情况下，众所周知，这些波浪是独一无二的，是张柳的作品。因此，我们的结果特别暗示，经典 Ostrovsky 方程的所有行波都是光谱稳定的。