The limit behavior of solutions for the Cauchy problem of the complex Ginzburg-Landau equation

This paper is devoted to the study of the limit behavior as epsilon down arrow 0 (or epsilon down arrow 0 and a down arrow 10) for the solutions of the Cauchy problem of the complex Ginzburg-Landau equation u(t) --> epsilonDeltau - iDeltau + (a + i) \u\(alpha)u = 0, u(0, x) = u(0)(x), 4/n less than or equal to alpha less than or equal to 4/(n - 2) (0 < alpha < 4/(n - 2) as epsilon down arrow 0 and a down arrow 0). We show that its solution will converge to the solution of the Cauchy problem for the semilinear Schrodinger equation v(t)-iDeltav+(a+i)\v\(alpha)v = 0, v(0,x) = u(0)(x) (a = 0 if epsilon down arrow 0 and a down arrow 0) in the spaces C(0, T; H-s) for any T > 0, s = 0, 1, and s(alpha) := n/2 - 2/alpha. Moreover, the sharp convergence rate in such spaces is also given. (C) 2002 John Wiley Sons, Inc.