Lower dimensional invariant tori with prescribed frequency for the nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut−uxx+mu+|u|2u+f(|u|2)u=0, subject to Dirichlet boundary conditions is considered, where the nonlinearity ff is a real analytic function near u=0u=0 with f(0)=f′(0)=0f(0)=f′(0)=0. It is proved that for each given constant potential mm and each prescribed integer b>1b>1, the above equation admits a Whitney smooth family of small-amplitude time quasi-periodic solutions, whose bb-dimensional frequencies are just small dilation of a prescribed Diophantine vector. Accordingly, we obtain the existence of lower dimensional invariant KAM tori with tangential frequencies constrained to a given Diophantine direction in an infinite-dimensional phase space setting.