利用 KdV-Calogero-Bogoyavlenskii-Schiff 方程研究海岸或湖泊环境中的浅水波
Investigation of shallow water waves near the coast or in lake environments via the KdV-Calogero-Bogoyavlenskii-Schiff equation
AI总结：
本文研究描述海岸及湖泊浅水波的 KdV-CBS 方程，基于二元 Bell 多项式获得双线性 Bäcklund 变换、Lax 对与无穷守恒律，并结合同宿测试和符号计算构造多类混合解。
摘要：
自然界中的浅水波现象吸引了学者们的关注，并在海啸、潮汐波、孤立波和水利工程等领域发挥重要作用。据此，针对各种自然环境中的浅水波现象，本文研究 KdV-Calogero-Bogoyavlenskii-Schiff（KdV-CBS）方程。基于二元 Bell 多项式理论，推导得到 KdV-CBS 方程的一种新的一般双线性 Bäcklund 变换、Lax 对和无穷守恒律，并证明其在 Lax 对意义下完全可积。通过结合同宿测试方法与符号计算，构造了多种类型的混合解。这些发现对该学科具有重要意义，为认识 KdV-CBS 方程的复杂动力学提供了重要洞见。我们希望研究结果能够帮助研究人员理解海洋、河流和沿海地区浅水波的非线性复杂现象。此外，本工作还可直接应用于其他非线性方程。
英文摘要：
Shallow water waves phenomena in nature attract the attention of scholars and play an important role in fields such as tsunamis, tidal waves, solitary waves, and hydraulic engineering. Hereby, fortheshallowwaterwavesphenomenainvariousnaturalenvironments, westudytheKdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation. Based on the binary Bell polynomial theory, a new general bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the KdV-CBS equation are derived, and it is proved that it is completely integrable in Lax pair sense. Various types of mixed solutions are constructed by using a combination of Homoclinic test method and symbolic computations. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of the KdV-CBS equation. We hope that our research results could help the researchers understand the nonlinear complex phenomena of the shallow water waves in oceans, rivers and coastal areas. Furthermore, the present work can be directly applied to other nonlinear equations.
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