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In this work, we mainly focus on the energy-supercritical nonlinearSchr\"odinger equation, $$  i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with$\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider theenergy-supercritical cases, that is, $p\in (\frac4{d-2},+\infty)$.  We prove that for radial initial data with high frequency, if it is outgoing(or incoming) and in rough space $H^{s_1}(\mathbb{R}^d)$ $(s_1<s_c)$ or itsFourier transform belongs to $W^{s_2,1}(\mathbb{R}^d)$ $(s_2<s_c)$, thecorresponding solution is global and scatters forward (or backward) in time. Wealso construct a class of large global and scattering solutions starting withmany bubbles, which are mingled with in the physical space and separate in thefrequency space. The analogous results are also valid for theenergy-subcritical cases.

Large global solutions for nonlinear Schr\"odinger equations III, energy-supercritical cases