The concept of a primitive potential for the Schrödinger operator on the line was introduced in [2,3,4]. Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this paper, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential.

In this paper we show that all algebro-geometric finite gap solutions to the Korteweg-de Vries equation can be realized as a limit of N-soliton solutions as N diverges to infinity (see remark 1 for the precise meaning of this statement). This is done using the the primitive solution framework initiated by [5,28,31]. One implication of this result is that the N-soliton solutions can approximate any bounded periodic solution to the Korteweg-de Vries equation arbitrarily well in the limit as N diverges to infinity. We also study primitive solutions numerically that have the same spectral properties as the algebrogeometric finite gap solutions but are not algebro-geometric solutions.

We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.

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.

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