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Hydrodynamic soliton.
Source: Christoph Finot, Kamal Hammani, Laboratoire Interdisciplinaire CARNOT de Bourgogne, UMR 5209
CNRS-Université de Bourgogne, Dijon, Bourgogne, France

 

In my research I use techniques of spectral and scattering theory to address:

  • Completely integrable models of dispersive waves in one and two space dimensions
  • Spectral geometry of compact and non-compact Riemannian manifolds
  • Spectral and scattering theory of Schrödinger operators

Most of my research over the past five years has concentrated on completely integrable, dispersive PDE. With Peter Miller, Catherine Sulem, and Jean-Claude Saut, I co-organized a three-week focus program on "Nonlinear dispersive PDE's and inverse scattering" at the Fields Institute in August 2017. A follow-up conference will take place at the Fields Institute May 21-24, 2019.

A seminal review paper in the area by Christian Klein and Jean-Claude Saut discusses the successes and challenges of applying PDE and inverse scattering methods to the study of nonlinear dispersive PDE. Among the key challenges in the area are:

  • Global well-posedness for rough initial data
  • Blow-up phenonmena for critical dispersive equations
  • Genericity of inverse scattering results for non-dispersive PDE
  • Semiclassical limits for dispersive PDE

Among my current research interests are:

  • Rigorous inverse scattering theory for the Intermediate Long Wave Equation
  • Nonlinear harmonic analysis of scattering maps
  • Global existence versus blow-up for the focussing Davey-Stewartson II equation
  • Global existence for solutions of the Novikov-Veselov equation