A Hamiltonian set-up for 4-layer density stratified Euler fluids

By means of the Hamiltonian approach to two-dimensional wave motions in heterogeneous fluids proposed by Benjamin, we derive a natural Hamiltonian structure for ideal fluids, density stratified in four homogenous layers, constrained in a channel of fixed total height and infinite lateral length. We derive the Hamiltonian and the equations of motion in the dispersionless long-wave limit, restricting ourselves to the so-called Boussinesq approximation. The existence of special symmetric solutions, which generalize to the four-layer case the ones obtained in the paper for the three-layer case, is examined.