Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms \(u^{2n+1}\) and \(u^{2m}u_{xx}\), the equation loses the conservation law \(E=\int _{{\mathbb {S}}} (u^2+u^2_x)\mathrm{d}x.\) This difficulty has been dealt with by establishing the energy inequality.

中文翻译：

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弱耗散浅水方程的破波现象

本文考虑的是一个弱耗散的浅水方程。这些参数取不同的值，其中包括几个不同的重要浅水方程，例如CH方程，DP方程，Novikov方程等。通过三种不同的方法研究了破波现象。由于存在高阶非线性项\（u ^ {2n + 1} \）和\（u ^ {2m} u_ {xx} \），该方程式失去了守恒定律\（E = \ int _ { {\ mathbb {S}}}（u ^ 2 + u ^ 2_x）\ mathrm {d} x。\）这个困难已经通过建立能量不平等来解决。