Abstract
In this paper, we consider the one-dimensional nonlinear Schrödinger equation i u t − u x x + m u + f ( | u | 2 ) u = 0 with periodic boundary conditions or Dirichlet boundary conditions, where f is a real analytic function in some neighborhood of the origin satisfying f ( 0 ) = 0 , f ′ ( 0 ) ≠ 0 . We prove that for each given constant potential m, when the frequencies, as a function of the amplitudes, can be regarded as the independent parameters, the equation admits a Whitney smooth family of small-amplitude, time almost-periodic solutions with all frequencies. The proof is based on a Birkhoff normal form reduction and an improved version of the KAM theorem.
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