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 ε ↓ 0 (or ε ↓ 0 and a ↓ 0) for the solutions of the Cauchy problem of the complex Ginzburg‐Landau equation u t − ε▵ u − i ▵ u + ( a + i)| u | α u = 0, u (0, x ) = u 0 ( x ), 4/ n ≤ α ≤ 4/( n − 2) (0 < α < 4/( n − 2) as ε ↓ 0 and a ↓ 0 ). We show that its solution will converge to the solution of the Cauchy problem for the semilinear Schrödinger equation v t − i▵ v + ( a + i)| v | α v = 0, v (0, x ) = u 0 ( x ) ( a = 0 if ε ↓ 0 and a ↓ 0) in the spaces C (0, T ; Ḣ s ) for any T > 0, s = 0, 1, and s (α) := n /2 − 2/α. Moreover, the sharp convergence rate in such spaces is also given. © 2002 John Wiley & Sons, Inc.