Exponential and algebraic decaying solitary waves and their connection to hydraulic fall solutions

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.