Travelling wave solutions for an internal wave model

In this work we consider a reduced system that models internal waves at the interface of two immiscible, inviscid, incompressible and irrotational fluids limited by a rigid lid at the top and a flat bottom.The system was previously derived by asymptotic expansions from the Euler equations for each layer and the complementary conditions that coupled them and consists of two integro-differential evolution equations for the interface wave profile and the upper layer averaged horizontal velocity.The aforementioned system is further reduced here to a single integro-differential evolution equation (the Regularized Benjamin equation) for the interface wave profile, in order to prove the existence of smooth periodic travelling wave solutions which capture the behaviour of internal waves observed in the ocean layers.