Solitary wave interactions of the GRLW equation

An approximate quasilinearization method for the solution of the generalized regularized long-wave (GRLW) equation based on the separation of the temporal and spatial derivatives, three-point, fourth-order accurate, compact difference equations, is presented. The method results in a system of linear equations with tridiagonal matrices, and is applied to determine the effects of the parameters of the GRLW equation and initial conditions on the formation of undular bores and interactions/collisions between two solitary waves. It is shown that the method preserves very accurately the first two invariants of the GRLW equation, the formation of secondary waves is a strong function of the amplitude and width of the initial Gaussian conditions, and the collision between two solitary waves is a strong function of the parameters that appear in the GRLW equation and the amplitude and speed of the initial conditions. It is also shown that the steepening of the leading and trailing waves may result in the formation of multiple secondary waves and/or an undular bore; the former interacts with the trailing solitary wave which may move parallel to or converge onto the leading solitary wave.