Long-time behavior of solution for coupled Ginzburg–Landau equations describing Bose–Einstein condensates and nonlinear optical waveguides and cavities

In this paper, the Cauchy problem of coupled Ginzburg–Landau (GL) equations describing Bose–Einstein condensates and nonlinear optical waveguides and cavities is considered. The long-time behavior of solution is investigated using a series of sharp a priori estimates in phase space E1 and E2, respectively. The global strong attractor in E2 is proved by energy equation method, and its bound of dimension is shown.