Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation
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In this paper, one-dimensional (1D) nonlinear Schrödinger equation [equation omitted] 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 Birkho ff normal form reduction and an improved KAM method.