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

In this work, the authors first show the existence of global attractors AεAε for the following lattice complex Ginzburg–Landau equation: View the MathML sourceiu̇m−(α−iε)(2um−um+1−um−1)+iκum+β|um|2σum=gm,m∈Z,ε>0, Turn MathJax on and A0A0 for the following lattice Schrödinger equation: View the MathML sourceiu̇m−α(2um−um+1−um−1)+iκum+β|um|2σum=gm,m∈Z. Turn MathJax on 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 View the MathML sourcelimε→0+distℓ2(Aε,A0)=0.