| Abstract: |
| In this talk, I will discuss a two-dimensional nonlinear dispersive PDE arising in the study of water waves called the Davey-Stewartson system (DS)
\begin{equation}
i\partial_{t}u+(\sigma\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})u = \mu|u|^{2}u +\beta\frac{\partial_{x_{1}}^{2}}{(\partial_{x_{1}}^{2}+\alpha\partial_{x_{2}}^{2})}(|u|^{2})u, \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^{2}, \enspace \sigma,\mu\in \{\pm 1\}, \alpha,\beta\in\mathbb{R},
\end{equation}
which for $\sigma=+1$ is formally similar to the $L^{2}$-critical cubic nonlinear Schr\{o}dinger equation (NLS) but differs by an additional nonlocal term. Specifically, I will discuss recent work on the global wellposedness and scattering for a particular case of DS with initial data in the critical $L^{2}$ space, which is inspired by Benjamin Dodson`s breakthrough work on the cubic NLS. Finally, I will discuss the question of the rigorous justification of DS as a multiple scales approximation for wave packet solutions to the water waves equation. |
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