07-55 Zhenguo Liang, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Quasi-Periodic Solutions for 1D Schr\"{o}dinger Equation with the Nonlinearity |u|^{2p}u (583K, pdf) Mar 7, 07

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Abstract. In this paper, one-dimensional (1D) nonlinear Schr\"{o}dinger equation $$iu_{t}-u_{xx}+|u|^{2p}u=0,\ p\in \N,$$ with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial normal form and scaling skills.

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