Applications of Complex Variables to Spectral Theory and Completely Integrable Partial Differential Equations

Abstract

We generalize the 1+1 Kaup--Broer system to an integrable 2+1 dimensional system via the dressing method. This allows us to compute N--soliton solutions to this 2+1 dimensional system, and also to the 1+1 Kaup--Broer system. This method also allows the computation of new solutions that generalize the N--soliton solutions a natural way. We formulate the inverse spectral problem for Hill's operators with bounded periodic potentials as a Riemann--Hilbert problem, and characterize the space of solutions to this Riemann--Hilbert problem. This gives an alternative proof that such a Hill's operator is determined uniquely by its spectral gaps, a Dirichlet eigenvalue in the closure of each spectral gap, and a signature for each Dirichlet eigenvalue in the interior of a spectral gap.