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.

翻译：本文研究色散管理非线性薛定谔（DM-NLS）方程 $$ i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d, $$ 以及非线性管理NLS（NM-NLS）方程： $$ i\partial_t u(t,x)+\Delta u(t,x)=\gamma(t)|u(t,x)|^{\frac4d}u(t,x), \quad x\in\R^d, $$ 其中 $\gamma(t)$ 为周期函数，在 $t\in (0,1]$ 时取值为 $-1$，在 $t\in (1,2]$ 时取值为 $1$。这两个模型的共同特征是聚焦与散焦效应周期性转换。对于经典聚焦NLS，已知初始数据 $$ u_0(x)=T^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4T} -i\frac{\omega^2}{T}}Q_\omega\left(\frac{x}{T}\right) $$ 会导出一个爆破解 $$(T-t)^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4(T-t)} -i\frac{\omega^2}{T-t}}Q_\omega\left(\frac{x}{T-t}\right), $$ 因此当 $T\leq1$ 时，该解同样是DM-NLS与NM-NLS的爆破解，并在第一个聚焦层发生爆破。对于DM-NLS，我们证明当 $T>1$ 时，上述初始数据 $u_0$ 不会导致有限时间爆破，且对应解是全局适定的。对于NM-NLS，我们证明了 $T\in(1,2)$ 时的全局适定性，并构造了可在任意聚焦层爆破的解。理论研究辅以广泛的数值探索，以理解两种模型中的稳定化效应并阐明其差异。