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.