CONJECTURES ABOUT PHASE TURBULENCE IN THE COMPLEX GINZBURG-LANDAU EQUATION

In the complex Ginzburg-Landau equation, we consider possible ''phase turbulent'' regimes, where asymptotic correlations are controlled by p hase fluctuations rather than by topological defects. Conjecturing tha t the decay of such correlations is governed by the Kardar-Parisi-Zhan g (KPZ) model of growing interfaces, we derive the following results: (1) A scaling ansatz implies that equal-time spatial correlations in 1 d, 2d, and 3d decay like e(-Ax2 zeta), where A is a nonuniversal const ant, and zeta=1/2 in 1d. (2) Temporal correlations decay as exp(-t(2 b eta)h(t/L(z))), with the scaling law <(beta)over bar> = <(zeta)over ba r>/z, where z = 3/2, 1.58..., and 1.66..., for d = 1,2, and 3 respecti vely. The scaling function h(y) approaches a constant as y --> 0, and behaves like y(2(beta-<(beta)over bar>)), for large y. If in 3d the as sociated KPZ model turns out to be in its weak-coupling (''smooth'') p hase, then, instead of the above behavior, the CGLE exhibits rotating long-range order whose connected correlations decay like 1/x in space or 1/t(1/2) in time. (3) For system sizes, L, and times t respectively less than a crossover length, L(c), and time, t(c), correlations are governed by the free-field or Edwards-Wilkinson (EW) equation, rather than the KPZ model. In 1d, we find that L(c) is large: L(c) similar to 35,000; for L < L(c) we show numerical evidence for stretched exponen tial decay of temporal correlations with an exponent consistent with t he EW value beta(EW)= 1/4.