河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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四维非线性薛定谔方程组解的整体适定性与散射

张茜，张晓岭，苌永行

河海大学数学学院，江苏南京 210098
DOI： 10.13763/j.cnki.jhebnu.nse.202601004

Global Well-posedness and Scattering for the Nonlinear Schrödinger Systemin Four Dimensions

ZHANG Xi,ZHANG Xiaoling,CHANG Yonghang

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摘要/Abstract

摘要：

非线性薛定谔方程组是量子力学和光学领域中重要的偏微分方程之一.研究了四维三次非线性薛定谔方程组解的整体适定性与散射行为.首先通过集中紧-刚性定理方法将方程组的解归结为几乎周期解，然后，借助长时间的Strichartz估计和频率局部化相互作用的Morawetz估计，排除了快速频率转化情况和准孤立子情况，进而证明出方程组具有整体适定性并且解散射.本研究不仅解决了四维薛定谔方程组这一关键问题，发展的频率局部化估计方法也为处理更高维或更复杂非线性项的系统提供了新思路.

Abstract：

The nonlinear Schrödinger equation is one of the important partial differential equations in the field of quantum mechanics and optics.This paper studies the global well-posedness and scattering behavior of solutions to the 4-dimensional cubic nonlinear Schrödinger system.First,the solution of the system of equations is reduced to an almost periodic solution by the method of concentration-compact-ness/rigidity theorem.Then,with the help of the long-term Strichartz estimation and the Morawetz estimation of the frequency localization interaction,the paper excludes the cases of fast frequency transition and quasi-soliton case,thereby proving the global well-posedness of the system and the scattering of its solutions.

关键词

关键词： 非线性薛定谔方程组；整体适定；散射；长时间Strichartz估计；Morawetz估计

Key words： the nonlinear Schrödinger equation ; global well-posedness ; scattering ; the long-term Strichartz estimation ; the Morawetz estimation

参考文献 12

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