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08-16【Chenmin Sun】管楼1418微分方程系列报告

澳门新葡新京:系统管理员发布时间:2019-08-09浏览次数:9

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TitleGIBBS MEASURE FOR THE FRACTIONAL NONLINEAR SCHRöDINGER EQUATIONS
SpeakerChenmin Sun    (Universite Cergy-Pointoise)
Time2019年8月16日            下午    16:00-17:00
Room东区管理科研楼   澳门新葡新京1418室

AbstractWe consider the fractional nonlinear Schr¨odinger equation with cubic nonlinearity: 
                                                 $$i/partial_tu-(-/partial_x^2)^{/alpha/2}u=|u|^2u./eqno(0.1)$$
(0.1) is a Hamiltonian system with conserved energy
                              $$H(u)=/int_{/mathbb{T}}/left({1/over 2}|D^{/alpha/2}u|^2+{1/over 4} |u|^4/right)dx.$$
The case $/alpha=2$ corresponds to the classical nonlinear Schr/¨odinger equation. I will first explain the construction of its Gibbs measure, which is formally of the form $d/mu=e^{-H(u)}du$, for the strong dispersive case $/alpha>1$. For the weak dispersive case $/alpha/leq 1$, a renormalization procedure is needed, in order to make sense of the formal expression. Next I will discuss three methods for construting global dynamics on the support of the Gibbs measure, according to the value of $/alpha$. This talk is based on a joint work with N. Tzvetkov.


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