Phase dynamics in the complex Ginzburg-Landau equation

J. Diff. Eqns. 199 (2004) 22-46

Ian Melbourne and Guido Schneider

Abstract

For ab > -1, stable time periodic solutions A(X,T)= A_q e^{iqX + i w_q T} $ are the locally preferred planform for the complex Ginzburg-Landau equation

A_T = A + (1+ia) A_XX - (1+ib) A|A|² .

In order to describe the spatial global behavior, an evolution equation for the local wave number q can be derived formally. The local wave number q satisfies approximately a conservation law

q_t = [h(q)]_x .

It is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation.