A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case
Published Web Locationhttps://doi.org/10.1016/j.jde.2014.12.023
In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.