Stability of Peaked Solitary Waves for a Class of Cubic Quasilinear Shallow-Water Equations
Stability of Peaked Solitary Waves for a Class of Cubic Quasilinear Shallow-Water Equations
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一类三次拟线性浅水方程的峰值孤立波稳定性
DOI：
10.1093/imrn/rnac032
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发表时间：
2022
影响因子：
1
通讯作者：
Liu, Yue
中科院分区：
文献类型：
--
作者：
Chen, Robin Ming;Di, Huafei;Liu, Yue
本文关注两类三次拟线性方程，它们可以作为从浅水近似到二维不可压缩欧拉方程的渐近模型导出。一类模型具有齐次三次非线性，包括可积的修正 Camassa-Holm (mCH) 方程和诺维科夫方程，另一类包含二次和三次非线性。这里证明这两个模型都具有局部峰值解。通过构造李雅普诺夫函数，这些峰值波在自然能量空间的小扰动下表现出动态稳定，且不受动量密度符号的限制。特别是，对于齐次三次非线性模型，我们能够进一步结合高阶守恒定律来得出轨道稳定性。我们的分析是基于守恒定律的大量使用、某些辅助函数的引入以及精确的连续性论证。
This paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.