Effects of convection on a modified GRLW equation

A time-linearization method that uses a Crank-Nicolson procedure in time and three-point, fourth-order accurate in space, compact difference equations, is presented and used to determine the solutions of the generalized regularized-long wave (GRLW) equation and a modified version thereof (mGRLW). The method results in a system of linear equations with block-tridiagonal matrices and provides the values of the solution and of its first- and second-order spatial derivatives. For the GRLW equation, it is shown that the initial conditions may result in the formation of a travelling wave and an almost stationary, small amplitude wave parallel to the left boundary, while the collision between two solitary waves may result in either radiation or the formation of an undular bore, but the solitary waves emerge from the interaction without a change in their shapes. It is shown that, after an initial transient behavior, the mGRLW equation may form an undular bore and a travelling wave whose speed is smaller than that of the GRLW equation. The collision between two solitary waves of the mGRLW equation may result in the disappearance of the initially leading wave and the formation of an undular bore. It is also shown that the propagation and interaction of solitary waves solutions of the GRLW and mGRLW equations depend strongly on the nonlinear and linear advection terms and the viscosity.