Under investigation in this paper is a higher order wave equation of the Korteweg–de Vries type, which describes the propagation of more complex wave in the shallow water. Bilinear form is derived based on the binary Bell polynomials. One-soliton solutions are constructed via the Hirota method. Two-soliton solutions are tried to be constructed, but it degenerates to the one-soliton solutions. Influences of the coefficients $\alpha$ and $\beta$ on the soliton velocity and amplitude are analyzed through the characteristic line.

中文翻译：

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高阶波动方程的双线性形式和孤子解

本文研究的是 Korteweg-de Vries 型的高阶波动方程，它描述了更复杂的波在浅水中的传播。双线性形式是基于二进制贝尔多项式导出的。通过 Hirota 方法构建单孤子解决方案。试图构造双孤子解，但它退化为一孤子解。系数的影响$\alpha$和$\beta$通过特征线对孤子速度和幅值进行分析。