二维倾斜粘性浅水方程的周期重力-毛细滚波解
Periodic gravity-capillary roll wave solutions to the inclined viscous shallow water equations in two dimensions
AI总结：
本文首次构造并研究了倾斜平面上二维粘性浅水系统的周期重力-毛细滚波解，通过隐函数定理与多参数分岔定理分类了产生小振幅解曲线的速度与倾角组合。
摘要：
我们研究了一个建立在倾斜平面上的粘性浅水系统的周期、二维、重力-毛细行波解。在将Reynolds数和Bond数视为固定且有限的同时，我们改变行波参考系的速度和倾斜角度，并识别出后两个参数的一个集合，该集合分类了哪些组合会产生非平凡的小振幅解曲线。我们的主要技术工具是隐函数定理和局部多参数分岔定理的结合。据作者所知，本文首次构造并数学研究了真正二维的粘性滚波例子。
英文摘要：
We study periodic, two-dimensional, gravity-capillary traveling wave solutions to a viscous shallow water system posed on an inclined plane. While thinking of the Reynolds and Bond numbers as fixed and finite, we vary the speed of the traveling frame and the degree of the incline and identify a set of the latter two parameters that classifies from which combinations nontrivial and small amplitude solution curves originate. Our principal technical tools are a combination of the implicit function theorem and a local multiparameter bifurcation theorem. To the best of the author's knowledge, this paper constitutes the first construction and mathematical study of properly two dimensional examples of viscous roll waves.
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