指数衰减与代数衰减孤立波及其与水力跌落解的联系
Exponential and algebraic decaying solitary waves and their connection to hydraulic fall solutions
AI总结：
本研究针对 fKdV 方程，揭示临界 Froude 数下指数衰减孤立波的非线性特征值属性及其向代数衰减孤立波连续谱的演化，并提出稳定数值方法构建水力跌落参数空间。
摘要：
受迫 Korteweg-de Vries（fKdV）方程描述了在任意地形上不可压缩、无粘性自由表面流动。我们研究了 fKdV 方程的孤立波解和水力跌落解。数值结果表明，在临界 Froude 数下计算指数衰减孤立波是一个非线性特征值问题。此外，我们展示了指数衰减孤立波如何演化为代数衰减孤立波的连续谱。一种利用波阻系数和桌面解的新颖稳定数值方法被用于生成水力跌落参数空间。我们展示了水力跌落解如何周期性地演化为指数衰减孤立波。
英文摘要：
The forced Korteweg-de Vries (fKdV) equation describes incompressible inviscid free surface flows over some arbitrary topography. We investigate solitary and hydraulic fall solutions to the fKdV equation. Numerical results show that the calculation of exponentially decaying solitary waves at the critical Froude number is a nonlinear eigenvalue problem. Furthermore we show how exponential decaying solitary waves evolve into the continuous spectrum of algebraic decaying solitary waves. A novel and stable numerical approach using the wave-resistance coefficient and tabletop solutions is used to generate the hydraulic fall parametric space. We show how hydraulic fall solutions periodically evolve into exponential decaying solitary waves.
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