CONVERGENCE OF SOLUTIONS OF THE BBM AND BBM-KP MODEL EQUATIONS

The Benjamin-Bona-Mahony (BBM) equation has proven to be a good approximation for the unidirectional propagation of small amplitude long waves in a channel where the crosswise variation can be safely ignored. The Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP) equation is the regularized version of the Kadomtsev-Petvia-shvili equation which arises in various modeling scenarios corresponding to nonlinear dispersive waves that propagate principally along the x -axis with weak dispersive effects undergone in the direction parallel to the y-axis and normal to the primary direction of propagation. There is much literature on mathematical studies regarding these well known equations, however the relationship between the solutions of their under-lying pure initial value problems is not fully understood. In this work, it is shown that the solution of the Cauchy problem for the BBM-KP equation converges to the solution of the Cauchy problem for the BBM equation in a suitable function space, provided that the initial data for both equations are close as the transverse variable y -> +/-infinity.