Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrodinger equation

The attractor of the weakly damped driven nonlinear Schrdinger equation in one spatial dimension with periodic boundary conditions is known to be smooth whenever the driving term is smooth (O. Goubet, Appl. Anal. 60 (1996), 99-119). We show that the attractor is in fact contained in a subclass of the real analytic functions provided the driving term is real analytic. Our method is based on a characterization of Gevrey classes (and thus of the real analytic functions) in terms of decay of Fourier coefficients. These tools were previously developed in the context of parabolic dissipative equations, and we extend their applicability to the case of weak damping. As a consequence of having an analytic attractor we prove that any two close enough points in the physical domain form a set of determining nodes for the long-time dynamics of the equation.

Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrodinger equation Academic Article