Abstract To describe large amplitude internal solitary waves in a two-layer system, we consider the high-order unidirectional (HOU) model that extends the Korteweg–de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model is valid only for weakly nonlinear waves, its coefficients depending on the depth and density ratios are adjusted such that the adjusted model can represent the main characteristics of large amplitude internal solitary waves, including effective wavelength, wave speed, and maximum wave amplitude. It is shown that the solitary wave solution of the adjusted HOU (aHOU) model agrees well with that of the strongly nonlinear Miyata–Choi–Camassa (MCC) model up to the maximum wave amplitude, which cannot be achieved by the original HOU model. To further validate the aHOU model, numerical solutions of the aHOU model are presented for the propagation and interaction of solitary waves and are shown to compare well with those of the MCC model. The aHOU model is further extended to the case of variable bottom and is solved numerically. In comparison with the MCC model for variable bottom, it is found that the aHOU model is a simple, but reliable theoretical model for large amplitude internal solitary waves, which would be useful for practical applications.

中文翻译：

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大振幅长内波调整系数的高阶单向模型

摘要 为了描述两层系统中的大振幅内部孤立波，我们考虑了高阶单向 (HOU) 模型，该模型扩展了具有高阶非线性和超前阶非线性色散的 Korteweg-de Vries 方程。虽然原始 HOU 模型仅对弱非线性波有效，但其依赖深度和密度比的系数进行了调整，使得调整后的模型可以表示大振幅内部孤立波的主要特征，包括有效波长、波速和最大值波幅。结果表明, 调整后的 HOU (aHOU) 模型的孤立波解与强非线性 Miyata-Choi-Camassa (MCC) 模型的孤立波解在最大波幅之前非常吻合, 这是原始 HOU 模型无法实现的。为了进一步验证 aHOU 模型，aHOU 模型的数值解被提出用于孤立波的传播和相互作用，并被证明与 MCC 模型的数值解可以很好地进行比较。aHOU 模型进一步扩展到变底的情况并进行了数值求解。与变底的MCC模型相比，发现aHOU模型是一种简单但可靠的大振幅内孤立波理论模型，有利于实际应用。