numerical stability of symmetric solitary-wave-like waves of a two-layer fluid—forced modified kdv equation

forced internal waves at the interface of a two-layer incompressible fluid in a two-dimensional domain with rigid horizontal boundaries are studied. the lower boundary is assumed to have a small obstruction. we derive a time-dependent forced modified kdv equation when the kdv theory fails and study the stabilities of four types of symmetric time-independent solitary-wave-like solutions numerically.