On blowup solution in NLS equation under dispersion or nonlinearity management

In this paper, we study the dispersion-managed nonlinear Schr\"odinger (DM-NLS) equation $$i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d,$$ and the nonlinearity-managed NLS (NM-NLS) equation: $$i\partial_t u(t,x)+\Delta u(t,x)=\gamma(t)|u(t,x)|^{\frac4d}u(t,x), \quad x\in\R^d,$$ where $\gamma(t)$ is a periodic function which is equal to $-1$ when $t\in (0,1]$ and is equal to $1$ when $t\in (1,2]$. The two models share the feature that the focusing and defocusing effects convert periodically. For the classical focusing NLS, it is known that the initial data $$u_0(x)=T^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4T} -i\frac{\omega^2}{T}}Q_\omega\left(\frac{x}{T}\right)$$ leads to a blowup solution $$(T-t)^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4(T-t)} -i\frac{\omega^2}{T-t}}Q_\omega\left(\frac{x}{T-t}\right),$$ so when $T\leq1$, this is also a blowup solution for DM-NLS and NM-NLS which blows up in the first focusing layer. For DM-NLS, we prove that when $T>1$, the initial data $u_0$ above does not lead to a finite-time blowup and the corresponding solution is globally well-posed. For NM-NLS, we prove the global well-posedness for $T\in(1,2)$ and we construct solution that can blow up at any focusing layer. The theoretical studies are complemented by extensive numerical explorations towards understanding the stabilization effects in the two models and addressing their difference.