Lie group analysis of nonlinear inviscid flows with a free surface under gravity

We applied the Lie group method in studying nonlinear inviscid flows with a free surface under gravity. This method reduces the number of independent variables by one. Therefore, for a system of partial differential equations with three independent variables we applied the method twice to yield a system represented by ordinary differential equations with appropriate corresponding conditions. We obtained analytical solutions for this system. Solutions for the free surface and the velocity components are obtained in closed form. The results are illustrated graphically for different parameters.