非線性薛丁格方程組的爆破解

Abstract

此篇論文裡,我們研究的是非線性薛丁格方程組的爆破解,方程組中的非 線性項係數為時間的函數。主要定理說明了此方程組能找到一組解 (Φ,Ψ)使得∥▽Φ∥L2(Ω)和∥▽Ψ∥L2(Ω)同時爆破,但我們並不清楚∥Φ∥L∞(Ω)和∥Ψ∥L∞(Ω)是否會同時爆破。

Here we study finite-time blow-up solutions of two-component system of nonlinear Schr¨odinger equations with a time-dependent nonlinearity. From main theorem in this paper, the system may have a solution (Φ,Ψ) such that both ∥▽Φ∥L2(Ω) and ∥▽Ψ∥L2(Ω) blow up at the same time but we don’t know whether ∥Φ∥L∞(Ω) and ∥Ψ∥L∞(Ω) blow up simultaneously.