Phase turbulence in the complex Ginzburg-Landau equation via Kuramoto-Sivashinsky phase dynamics

We study the Complex Ginzburg-Landau initial value problem partial derivative(t)u = (1 + ialpha) partial derivative(x)(2)u + u - (1 + ibeta) u\u\(2), u(x, 0) = u(0)(x), (CGL) for a complex field u is an element ofC, with alpha, beta is an element ofR. We consider the Benjamin-Feir linear instability region 1+alphabeta = -epsilon(2) with epsilon << 1 and alpha(2) < 1/2. We show that for all epsilon <= O (root 1-2 alpha(2) L-0(-32/37)), and for all initial data u(0) sufficiently close to 1 (up to a global phase factor e(iphi0),phi(0)is an element ofR) in the appropriate space, there exists a unique (spatially) periodic solution of space period L-0. These solutions are small even perturbations of the traveling wave solution, u = (1+alpha(2) s) e(i phi0-ibetat) e(ialpha eta), and s,eta have bounded norms in various L-p and Sobolev spaces. We prove that s approximate to 1/2 eta apart from O(epsilon(2)) corrections whenever the initial data satisfy this condition, and that in the linear instability range L-0(-1) less than or equal to epsilon less than or equal to O(L-0(-32/37)), the dynamics is essentially determined by the motion of the phase alone, and so exhibits 'phase turbulence'. Indeed, we prove that the phase eta satisfies the Kuramoto-Sivashinsky equation partial derivative(t)eta = -(1+alpha(2)/2) Delta(2)eta - epsilon(2)Deltaeta - (1 +alpha(2)) (eta')(2) (KS) for times t(0) < O(epsilon(-52/5) L-0(-32/5)), while the amplitude 1+alpha(2)s is essentially constant.