We successfully derived some exact solitary wave solutions of the nonlinear approximate equations with long waves in shallow water by using the hyperbola function method.

用双曲函数法，成功得到了非线性浅水长波近似方程组的精确孤立波解。

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Therefore, the study of higher-order nonlinear approximate to shallow water waves enriches the water wave dynamics and is crucial to engineering practice.

因此，研究非线性浅水波的高阶渐近解对丰富近海水波动力学理论和指导工程实践均具有重要意义。

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Therefore, the study of higher-order nonlinear approximate to shallow water waves enriches the water wave dynamics and is crucial to engineering practice.

因此，研究非线性浅水波的高阶渐近解对丰富近海水波动力学理论和指导工程实践均具有重要意义。

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