  We consider a model, which we named the Kakinuma model, for interfacial
gravity waves. As is well-known, the full model for interfacial gravity waves
has a variational structure whose Lagrangian is an extension of Luke's
Lagrangian for surface gravity waves, that is, water waves. The Kakinuma model
is a system of Euler-Lagrange equations for approximate Lagrangians, which are
obtained by approximating the velocity potentials in the Lagrangian for the
full model. In this paper, we first analyze the linear dispersion relation for
the Kakinuma model and show that the dispersion curves highly fit that of the
full model in the shallow water regime. We then analyze the linearized
equations around constant states and derive a stability condition, which is
satisfied for small initial data when the denser water is below the lighter
water. We show that the initial value problem is in fact well-posed locally in
time in Sobolev spaces under the stability condition, the non-cavitation
assumption and intrinsic compatibility conditions in spite of the fact that the
initial value problem for the full model does not have any stability domain so
that its initial value problem is ill-posed in Sobolev spaces. Moreover, it is
shown that the Kakinuma model enjoys a Hamiltonian structure and has
conservative quantities: mass, total energy, and in the case of the flat
bottom, momentum.
