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)
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.