C03 – Nonlinear dispersive equations and nonlinear Fourier analysis
Nonlinear waves in diverse areas such as nonlinear optics, large quantum systems and water waves are described asymptotically by dispersive equations, with the cubic nonlinear Schrödinger equation
being perhaps the most prominent among them. This equation, as well as a host of similar dispersive equations, can be solved by the inverse scattering method, which is based on a nonlinear variant of the Fourier transform. The relevance of such integrable equations hinges on a certain robustness: Despite the nonlinearity of the equation, large and highly complex solutions are expected to consist of building blocks described by asymptotic dispersive equations. The goals of this project are threefold.
- We study solutions to dispersive equations with a focus on harmonic analysis, in particular the robustness of the flow of energy, (scaling) critical regimes and initial data and nonlinear estimates in critical spaces.
- We pursue foundational research on elementary properties of the nonlinear Fourier transform as a tool for anticipated future applications to dispersive equations.
- We consider an analogue of Landau damping near stable shear flows for the Euler equation.
The configurations under study are composed of a large number of wave packets with controlled interactions, and they may be among the most natural ’high dimensional’ nonlinear models. The difference between the well understood formal linearization by the nonlinear Fourier transform and the analytic challenges for large nondecaying data and the nonlinear Fourier transform are at the center of the project.