Grimshaw derived the rotation modified Kadomtsev-Petviashivili Equation (RKP Eq.) to discribe long surface or internal waves in the presence of rotation. The initial-boundary value problems of RKP equation are studied numerically in this paper. It is shown that solitary-like waves prapogating to the right can be found, whose amplitudes decay in the direction perpendicular to the direction of prapogation and the wave fronts are curvedback. The solitary waves remain unsteady and are always accompanied by Poincare waves travelling to the left.These effects are more noticeable as the effects of rotation are increased.

This Paper deals with the initial value problems of the Modified KdV(MKdV) equation. First, the analytic solutions of solitary wave and interaction of two solitons are obtained by the Bäcklund Transformation. Then, a numerical scheme of a kind of Petrov-Galerkin finite element methods is derived and used to solve the initial value problems of MKdV equation numerically. Finally, the comparison of numerical and analytic solutions is made and shows the excellent accurancy and stability of the numerical scheme used in this paper.

A numerical solution of the initial value problem of regularized long-Wave equation (RLW Equation) was made by a Petrov-Galerkin finite elemnte method. The numerical resuts are consistent with the exact solution of propagation of single solitary wave, due to second-order and fourth-order accuracy for the time and spacing variable respectively. For the collision of two solitons, there is a slight oscilatoiy wave trail after their collision, so it doesn't have the quality of soliton strictly.