  In this paper a three-parameter family of Boussinesq systems is studied. The
systems have been proposed as models of the propagation of long internal waves
along the interface of a two-layer system of fluids with rigid-lid condition
for the upper layer and under a Boussinesq regime for the flow in both layers.
We first present some theoretical properties of well-posedness, conservation
laws and Hamiltonian structure of the systems. Then the corresponding periodic
initial-value problem is discretized in space by the spectral Fourier Galerkin
method and for each system, error estimates for the semidiscrete approximation
are proved. The rest of the paper is concerned with the study of existence and
the numerical simulation of some issues of the dynamics of solitary-wave
solutions. Standard theories are used to derive several results of existence of
classical and generalized solitary waves, depending on the parameters of the
models. A numerical procedure is used to generate numerically approximations of
solitary waves. These are taken as initial conditions in a computational study
of the dynamics of the solitary waves, both classical and generalized. To this
end, the spectral semidiscretizations of the periodic initial-value problem for
the systems are numerically integrated by a fourth-order
Runge-Kutta-composition method based on the implicit midpoint rule. The fully
discrete scheme is then used to approximate the evolution of perturbations of
computed solitary wave profiles, and to study computationally the collisions of
solitary waves as well as the resolution of initial data into trains of
solitary waves.
