Solutions to the Kaup-Broer system and its (2+1) dimensional integrable generalization via the dressing method

In this paper we formulate the nonlocal dbar problem dressing method of Manakov and Zakharov (Zakharov and Manakov, 1984, 1985; Zakharov, 1989) for the 4 scaling classes of the (1+1) dimensional Kaup-Broer system (Broer, 1975; Kaup, 1975). The applications of 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 Rogers and Pashaev (2011). This method allows computation of solutions to all scaling classes of the Kaup-Broer system. We then consider the case of non-capillary waves with 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 analogous to the solutions derived in (Dyachenko and Zakharov, 2016; Zakharov and Dyachenko, 2016; Zakharov et al., 2016) for the KdV equation. We derive dressing functions for finite gap solutions, and compute counter propagating dispersive shockwave type solutions numerically. Published by Elsevier B.V.