Conference and Workshop: Scattering and Inverse Scattering in Multidimensions

Grants and Contracts Details


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 large-N 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 multi-dimensional 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. *One-dimensional limits. (2+1)-dimensional dispersive equations such as the DS-II equation are paired with one-dimensional partners (the NLS equation in the case of DS-II) 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 one-dimensional 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.
Effective start/end date3/15/142/28/15


  • National Science Foundation: $28,726.00


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