  In this paper we formulate the nonlocal dbar problem dressing method of
Manakov and Zakharov [28, 29, 27] for the 4 scaling classes of the 1+1
dimensional Kaup--Broer system [7, 13]. The method for the 1+1 dimensional
Kaup--Broer systems are reductions of a method for a complex valued 2+1
dimensional completely integrable partial differential equation first
introduced in [23]. This method allows computation of solutions to all cases of
the Kaup--Broer system. We then consider the case of non-capillary waves with
usual gravitational forcing, and use the dressing method to compute N-soliton
solutions and more general solutions in the closure of the N-soliton solutions
in the topology of uniform convergence in compact sets called primitive
solutions. These more general solutions are an analogue of the solutions
derived in [11, 30, 31] for the KdV equation. We derive dressing functions for
finite gap solutions. We compute counter propagating dispersive shockwave type
solutions numerically.
