MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION SMALL DATA SOLUTION ON PRODUCT SPACE.

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schrödinger equation (NLS) posed on the spatial domain ℝ X 핋d, 1 ≤ d ≤ 4. We first prove the local well-posedness in C(I; L²xHsy} ⋂ C (I ; LXx,y) for solutions with initial data U0 ∈ Hx0,1 L²y ⋂ L²x Hsy. Then, for sufficiently small, smooth, decaying data, we prove global existence and derive modified asymptotic dynamics by using the wave packet method and normal form corrections. The modified scattering behavior on ℝ X 핋d combines the modified scattering of the cubic NLS on real line R with cubic NLS dynamics on the torus. We also consider the corresponding asymptotic completeness problem. [ABSTRACT FROM AUTHOR]