Georgia Institute of Technology College of Sciences

In this talk, I'll discuss the asymptotics of the cubic nonlinear Schrödinger equation with potential in dimension 1 for small, localized initial data. In the case when the potential is equal to 0, it has been known for some time that solutions exhibit modified scattering. Due to additional complications introduced by the potential, the case with V nonzero has not been addressed until recently. 

Here, we present a method to obtain asymptotics for this problem.  The main ingredients are  (1) a new linear identity, which allows us to relate certain vector field-like quantities for the problem with a potential to those for the problem with no potential, and (2) an adaptation of the method of testing with wave packets introduced by Ifrim and Tataru. Compared to previous results, this method can handle potentials with slower decay at infinity.

Mean-field games (MFG) theory is a mathematical framework for studying large systems of agents who play differential games. In the PDE form, MFG reduces to a Hamilton-Jacobi equation coupled with a continuity or Kolmogorov-Fokker-Planck equation. Theoretical analysis and computational methods for these systems are challenging due to the absence of strong regularizing mechanisms and coupling between two nonlinear PDE.

One approach that proved successful from both theoretical and computational perspectives is the variational approach, which interprets the PDE system as KKT conditions for suitable convex energy. MFG systems that admit such representations are called potential systems and are closely related to the dynamic formulation of the optimal transportation problem due to Benamou-Brenier. Unfortunately, not all MFG systems are potential systems, limiting the scope of their applications.

I will present a new approach to tackle non-potential systems by providing a suitable interpretation of the Benamou-Brenier approach in terms of monotone inclusions. In particular, I will present advances on the discrete level and numerical analysis and discuss prospects for the PDE analysis.

While the research on water waves modeled by Euler's equations has a long history, mainly in the last two decades traveling periodic rotational waves have been constructed rigorously by means of bifurcation theorems. After introducing the problem, I will present a new reformulation in two dimensions in the pure-gravity case, where the problem is equivalently cast into the form "identity plus compact," which is amenable to Rabinowitz's global bifurcation theorem. The main advantages (and the novelty) of this new reformulation are that no simplifying restrictions on the geometry of the surface profile and no simplifying assumptions on the vorticity distribution (and thus no assumptions regarding the absence of stagnation points or critical layers) have to be made. Within the scope of this new formulation, global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. Moreover, I will discuss the possible alternatives for the global set of solutions, as well as their nodal properties. This is joint work with Erik Wahlén.

We will discuss time-dependent, nonlinear “inverse scattering” in the setting of nonlinear Schrödinger equations.  In particular, we will show that it is possible to recover an unknown nonlinearity from the small-data scattering behavior of solutions.  Time permitting, we will also discuss stability estimates for reconstruction, as well as recovery from modified scattering behavior.  This talk will include some joint work with R. Killip and M. Visan, as well as with G. Chen.