The finite dimension behavior of the Schrödinger equation with nonlocal integral nonlinearity

Abstract

In this paper we discuss the long time behavior of the initial problem $$iu_t + u_{xx} + \beta \left| u \right|^p u + \gamma u\int_{ - \infty }^x {\left| u \right|^2 dx + iku = g,} $$ (1.1) $$u\left| {_{t = 0} } \right. = u^0 .$$ (1.2) We show that in a weighted Hilbert space there, exists a global attractor which is weakly compact and has finite Hausdorff and fractal dimension.