Non-linear Smoothing for the Periodic Generalized Non-linear Schrödinger Equation

Abstract

We consider the periodic non-linear Schrödinger equation with non-linearity given by $|u{|}^{p-1}u$ for odd $p>1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the linear evolution is in a smoother space. We then study forced and damped defocusing non-linear Schrödinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smootheness for global attractors in the energy space.

22 pages