Boyang Su, USC

Title: Global solutions for systems of quadratic nonlinear Schrödinger equations in 3D

Abstract: The existence of global solutions for the Schrödinger equation
\begin{equation}
     i\partial_t u + \Delta u = P(u),
     \tag{NLS}
\end{equation}
where $u(t,x):\mathbb{R}\times\mathbb{R}^n\rightarrow \mathbb{C}$ and $P$ is a nonlinear homogeneous function of degree $d$, has been extensively studied.  Much work focused on gauge-invariant nonlinearities of the form $P(u)=\lambda|u|^{d-1}u$, where the solutions satisfy several conservation laws. However, the problem becomes more complicated as we consider a general homogeneous $P$. Especially for smaller $d$, as the dispersive effect becomes weaker. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making 3D NLS with quadratic nonlinearity particularly interesting.

In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized initial data. To tackle the challenge posed by the $u\Bar{u}$-type nonlinearity, we introduce an \textquotedblleft$\epsilon$\textquotedblright regularization to the lower frequency part of the nonlinear term.