Grants and Contracts per year
Grants and Contracts Details
Description
Overview:
The purpose of this conference and workshop is to enable dramatic breakthroughs in the theory
of integrable nonlinear wave equations in (2+1) dimensions and related problems. To accomplish
this, we will bring together established researchers in three distinct areas: inverse problems,
integrable systems, and nonlinear dispersive wave equations. We also will involve postdocs
and students through a series of tutorial lectures and the workshop component of the meeting.
Intellectual Merit :
The conference and workshop will focus on the following open problems:
*Semiclassical limits. We will study semiclassical limits of direct and inverse scattering
transforms for (2+1) dimensional problems.
*Eigenvalue distributions of random normal matrices. The analytical problem of largeN asymptotic
behavior of random normal N x N matrices has many features in common with semiclassical limits
for dispersive nonlinear wave equations. We seek to develop new methods for this problem that
will impact simultaneously the fields of approximation theory (via new results on orthogonal
polynomials), mathematical physics (via new results on eigenvalue statistics), and integrable
nonlinear waves.
*Direct/Inverse scattering and exceptional sets. Nonuniqueness in the multidimensional scattering
problems that define the scattering data for integrable equations in two space dimensions
leads to singularities in the data and to such interesting
physical phenomena as "lump" solitons, "line" solitons, and solutions of the Cauchy problem
that blow up in finite time. We seek to clarify and classify the possible forms of the exceptional
sets of points where the scattering transform may have singularities and determine the solutions
they induce.
*Onedimensional limits. (2+1)dimensional dispersive equations such as the DSII equation
are paired with onedimensional partners (the NLS equation in the case of DSII) to which
they reduce for solutions independent of one of the spatial variables. We seek to
understand the relationship between inverse scattering in one and two dimensions in order
to elucidate such phenomena in two dimensions as the line solitons which arise from onedimensional
soliton solutions.
Broader Impacts :
Expertise in three different areas of applied mathematics  inverse problems, integrable
systems, and nonlinear dispersive wave equations  is required to make progress on the questions
addressed here. The purpose of this conference is to forge new collaborative links needed
between researchers from these different groups. It will be crucial going forward for some
representatives of the next generation of researchers (i.e., students and postdocs) to experience
this nucleation of ideas common to three areas in order to carry the project to the next stages.
This conference is intended to give students and postdocs the interdisciplinary expertise
and perspective to be successful. The conference will also initiate synergistic collaborations
that will produce greater research progress than could be achieved separately in the three
key areas.
Status  Finished 

Effective start/end date  3/15/14 → 2/28/15 
Funding
 National Science Foundation: $28,726.00
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Projects
 1 Finished

Restricted Scope: Participant Support: Conference and Workshop: Scattering and Inverse Scattering in Multidimensions
3/15/14 → 2/28/15
Project: Research project