  Considered here are two systems of equations modeling the two-way propagation
of long-crested, long-wavelength internal waves along the interface of a
two-layer system of fluids in the Benjamin-Ono and the Intermediate Long-Wave
regime, respectively. These systems were previously shown to have solitary-wave
solutions, decaying to zero algebraically for the Benjamin-Ono system, and
exponentially in the Intermediate Long-Wave regime. Several methods to
approximate solitary-wave profiles were introduced and analyzed by the authors
in Part I of this project. A natural continuation of this previous work,
pursued here, is to study the dynamics of the solitary-wave solutions of these
systems. This will be done by computational means using a discretization of the
periodic initial-value problem. The numerical method used here is a Fourier
spectral method for the spatial approximation coupled with a fourth-order,
explicit Runge-Kutta time stepping. The resulting, fully discrete scheme is
used to study computationally the stability of the solitary waves under small
and large perturbations, the collisions of solitary waves, the resolution of
initial data into trains of solitary waves, and the formation of dispersive
shock waves. Comparisons with related unidirectional models are also
undertaken.
