Applications of Complex Variables to Spectral Theory and Completely Integrable Partial Differential Equations
AuthorNabelek, Patrik Vaclav
AdvisorZakharov, Vladimir E.
McLaughlin, Kenneth D. T.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractWe 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.
Degree ProgramGraduate College