Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors AεAε for the following lattice complex Ginzburg–Landau equation: iu̇m−(α−iε)(2um−um+1−um−1)+iκum+β|um|2σum=gm,m∈Z,ε>0, and A0A0 for the following lattice Schrödinger equation: iu̇m−α(2um−um+1−um−1)+iκum+β|um|2σum=gm,m∈Z. Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+ε→0+. Also they prove the upper semicontinuity of AεAε as ε→0+ε→0+ in the sense that limε→0+distℓ2(Aε,A0)=0.