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Canonical system of equations for 1D water waves
Studies in Applied Mathematics ( IF 2.6 ) Pub Date : 2020-03-02 , DOI: 10.1111/sapm.12305
Alexander I. Dyachenko 1, 2
Affiliation  

One of the essential tasks of the theory of water waves is a construction of simplified mathematical models, which are applied to the description of complex events, such as wave breaking, appearing of freak waves in the assumption of weak nonlinearity. The Zakharov equation and its simplification, such as nonlinear Schrodinger equations and Dysthe equations, are among them. Recently, for unidirectional waves, the so‐called super compact equation was derived in Ref. 1. In the present article, the waves moving in both directions are considered. Namely, the waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four‐wave interactions of water waves allows describing the evolution of these two groups as a system of two equations. One of the significant consequences of this decomposition is the conservation of the number of waves in each particular group. To derive this system of equations, a particular canonical transformation is used. This transformation is possible due to the miraculous cancellation of the four‐wave interaction for some groups of waves in the one‐dimensional wave field. The obtained equations are remarkably simple. They can be called a canonical system of equations. They include a nonlinear wave term together with an advection term that can describe the initial stage of wave‐breaking. They also include interaction terms (between counter‐streaming waves). It is also suitable for analytical study as well as for numerical simulation.

中文翻译:

一维水波方程的典范系统

水波理论的基本任务之一是构造简化的数学模型,该模型可用于描述复杂事件,例如,在弱非线性的假设下,破波,畸形波的出现。Zakharov方程及其简化形式,例如非线性Schrodinger方程和Dysthe方程,都在其中。最近,对于单向波,参考文献中推导了所谓的超紧凑方程。1.在本文中,考虑了在两个方向上移动的波。即,二维深水自由表面上的波浪可以分为两组:向右移动的波浪和向左移动的波浪。水波的四波相互作用的特定特征允许将这两组的演化描述为两个方程组。这种分解的重大后果之一是每个特定组中波数的守恒。为了推导该方程组,使用了特定的规范变换。由于一维波场中某些波组奇迹般地消除了四波相互作用,因此这种转换是可能的。所获得的方程式非常简单。它们可以被称为方程的规范系统。它们包括一个非线性波浪项和一个对流项,可以描述波浪破碎的初始阶段。它们还包括交互作用项(在逆流波之间)。它也适用于分析研究以及数值模拟。为了推导该方程组,使用了特定的规范变换。由于一维波场中某些波组奇迹般地消除了四波相互作用,因此这种转换是可能的。所获得的方程式非常简单。他们可以被称为方程的规范系统。它们包括一个非线性波浪项和一个对流项,可以描述波浪破碎的初始阶段。它们还包括交互作用项(在逆流波之间)。它也适用于分析研究以及数值模拟。为了推导该方程组,使用了特定的规范变换。由于一维波场中某些波组奇迹般地消除了四波相互作用,因此这种转换是可能的。所获得的方程式非常简单。它们可以被称为方程的规范系统。它们包括一个非线性波浪项和一个对流项,可以描述波浪破碎的初始阶段。它们还包括交互作用项(在逆流波之间)。它也适用于分析研究以及数值模拟。所获得的方程式非常简单。他们可以被称为方程的规范系统。它们包括一个非线性波浪项以及一个可以描述波浪破碎初始阶段的对流项。它们还包括交互作用项(在逆流波之间)。它也适用于分析研究以及数值模拟。所获得的方程式非常简单。他们可以被称为方程的规范系统。它们包括一个非线性波浪项和一个对流项,可以描述波浪破碎的初始阶段。它们还包括交互作用项(在逆流波之间)。它也适用于分析研究以及数值模拟。
更新日期:2020-03-02
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