Dispersive blow-up for nonlinear Schrödinger equations revisited
Dispersive blow-up for nonlinear Schrödinger equations revisited
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重新审视非线性薛定谔方程的色散爆炸
DOI：
10.1016/j.matpur.2014.02.006
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Christof Sparber
中科院分区：
文献类型：
--
作者：
J. Bona;J. Saut;G. Ponce;Christof Sparber
本文重新讨论了非线性薛定谔方程有限时间色散爆破的可能性。这种数学现象是海洋和光学流氓波的可能解释之一。在一维非线性薛定谔方程中，色散爆破的事实已经在文献[9]中出现。在目前的工作中，现有的结果在几个方面进行了推广。在一个方向上，理论被扩展到包括Davey-Stewartson和Gross-Pitaevskii方程。在另一个，色散爆破显示，以获得非线性薛定谔方程的空间维度大于一个，更一般的幂律非线性。作为我们分析的一个副产品，我们得到了Duhamel公式中积分项的一个尖锐的整体光滑估计。
The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.