In the present study we describe the asymptotic perturbation method to derive a two-dimensional nonlocal shallow-water model equation in the context of full water waves. Starting from the incompressible and irrotational governing equations in the three-dimensional water waves, we show that such a equation arises in the modeling of the propagation of shallow water waves over a flat bed. The resulting equation is a two dimensional Camassa-Holm equation-type with weakly transverse effect for the horizontal velocity component. The equation captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg-de Vries and Kadomtsev-Petviashvili equations. We also address some properties of this model equation and how it relates to the surface wave. Analytically, we establish the local well-posedness of this model in a suitable Sobolev space. We then investigate formation of singularities and existence of traveling-wave solutions to this quasi-linear model equation with an emphasis on the understanding of weak transverse effect. Finally, we provide a Liouville-type property to obtain unique continuation result for the strong solution.

在本研究中，我们描述了在全水波背景下推导二维非局部浅水模型方程的渐近微扰方法。从三维水波中的不可压缩和无旋控制方程开始，我们表明这样的方程出现在浅水波在平床上传播的建模中。由此产生的方程是一个二维 Camassa-Holm 方程类型，对水平速度分量具有弱横向效应。该方程比经典色散可积方程（如 Korteweg-de Vries 和 Kadomtsev-Petviashvili 方程）捕捉到更强的非线性效应。我们还讨论了这个模型方程的一些属性以及它与表面波的关系。分析地，我们在合适的 Sobolev 空间中建立该模型的局部适定性。然后，我们研究奇点的形成和该准线性模型方程的行波解的存在，重点是对弱横向效应的理解。最后，我们提供了一个 Liouville 类型的性质，以获得强解的唯一延续结果。