The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reflectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = [special characters omitted] in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors.
In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q( x, 0) = qχ[−L,L ] for real amplitudes q. Using Riemann-Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an [special characters omitted](1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.
|Commitee:||Ercolani, Nick, Flaschka, Hermann|
|School:||The University of Arizona|
|School Location:||United States -- Arizona|
|Source:||DAI-B 70/08, Dissertation Abstracts International|
|Keywords:||NLS, Nonlinear waves, Schrodinger equation, Square barrier|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be