Torrential forced KdV equation: soliton solutions over a hole

Torrential flow of an incompressible and a perfect fluid over a hole in infinite channel is investigated. The flow is supposed to be two-dimensional, steady and irrotational. The surface tension is neglected but the gravity is included in the dynamic boundary condition. The issue has been programed and run on a computer to solve a nonlinear ordinary differential equation of the third order, called the forced Korteweg de Vries equation by using the Newton’s method. The obtained results in the torrential flow have been represented graphically by varying the shape of the hole and its depth, also by giving a different values of the Froude number λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}.