Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity

We prove existence of small-amplitude solitary and generalized solitary gravity waves traveling at the interface and the free surface of a two-dimensional two-layer fluid with vorticity and finite thickness. In both layers of the fluid the horizontal velocity of the fluid particle does not exceed the wave speed throughout the domain. The arguments are based on a formulation of the hydrodynamic problem as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A center-manifold reduction technique and a variety of dynamical systems methods are employed to detect homoclinic solutions to 0 and homoclinic solutions to periodic orbits to the reduced system, which correspond respectively to solitary and generalized solitary water waves.