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 took 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. See also their recent monograph Nonlinear dispersive equations--inverse scattering and PDE methods (Springer, Applied Mathematical Sciences, vol. 209). Among the key challenges in the area are:

• Global well-posedness for rough initial data
• Blow-up phenomena 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 for solutions of the Novikov-Veselov equation
• Long-time asymptotics for solutions of the KP I equation