In this paper, one-dimensional (1D) nonlinear Schrödinger equationi ut - ux x + m u + | u |4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. © 2006 Elsevier Inc. All rights reserved.
Geng, J., & Yi, Y. (2007). Quasi-periodic solutions in a nonlinear Schrödinger equation. Journal of Differential Equations, 233(2), 512–542. https://doi.org/10.1016/j.jde.2006.07.027