Crabb M, Akhmediev N
Department of Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT, 2600, Australia.
Phys Rev E. 2021 Feb;103(2-1):022216. doi: 10.1103/PhysRevE.103.022216.
Using Levi-Cività's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity approximations for shallow water. The complex KdV equation describes the nontrivial dynamics of all water particles from the surface to the bottom of the water layer. A crucial step made in our work is the proof that a natural consequence of the complex KdV theory is that the wave elevation is described by the real KdV equation. The complex KdV approach in the theory of shallow fluids is thus more fundamental than the one based on the real KdV equation. We demonstrate how it allows direct calculation of the particle trajectories at any point of the fluid, and that these results agree well with numerical simulations of other authors.
利用列维-奇维塔理想流体理论，我们推导出了复科特韦格-德弗里斯（KdV）方程，该方程描述了浅流体的复速度，精度可达一阶。我们使用摄动理论以及浅水的长波、慢变速度近似。复KdV方程描述了从水层表面到水底所有水粒子的非平凡动力学。我们工作中一个关键步骤是证明复KdV理论的一个自然结果是波高由实KdV方程描述。因此，浅流体理论中的复KdV方法比基于实KdV方程的方法更具基础性。我们展示了它如何允许直接计算流体中任意点的粒子轨迹，并且这些结果与其他作者的数值模拟结果吻合得很好。
