Homogenization for the cubic nonlinear Schrödinger equation on ℝ²

Abstract: The cubic nonlinear Schr\"odinger equation on $\mathbb R^2$ is given by $$i \partial_t u +\Delta u=\bar g |u|^2 u, \quad u(0)=u_0 \in L^2(\mathbb R^2).$$ This equation comes in two flavors, depending on the sign of $\bar g$: When $\bar g<0$, the self-interaction described by the nonlinearity is attractive. Heuristiaclly, the nonlinear part is working to counteract the dispersive effects of the linear part; indeed, finite time blow-up is possible. On the other hand, the case $\bar g>0$ indicates a repulsive self-interaction. In this regime the question of well-posedness for general initial data in $L^2$ was a long-standing problem in the field until its recent resolution by Dodson.

In this talk we will look at the corresponding inhomogeneous problem $$i \partial_t u +\Delta u=g(nx) |u|^2 u$$ with initial data in $L^2$, where $g$ does not necessarily have a fixed sign. We will discuss how it relates to the homogeneous NLS above and derive sufficient conditions on $g$ to ensure existence and uniqueness of global solutions for $n$ large, as well as homogenization.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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