Conjectures about phase turbulence in the complex Ginzburg-Landau equation

Abstract

In the complex Ginzburg-Landau equation, we consider possible "phase turbulent" regimes, where asymptotic correlations are controlled by phase fluctuations rather than by topological defects. Conjecturing that the decay of such correlations is governed by the Kardar-Parisi-Zhang (KPZ) model of growing interfaces, we derive the following results: (1) A scaling ansatz implies that equal-time spatial correlations in 1d, 2d, and 3d decay like e^-Ax2ζ, where A is a nonuniversal constant, and ζ = ½ in 1d. (2) Temporal correlations decay as exp(-t^2βh(t/L^z)), with the scaling law β = ζ/z, where z = 3/2, 1.58 …, and 1.66 …, for d = 1, 2 and 3 respectively. The scaling function h(y) approaches a constant as y→0, and behaves like y^2(β-β) for large y. If in 3d the associated KPZ model turns out to be in its weak-coupling ("smooth") phase, then, instead of the above behavior, the CGLE exhibits rotating long-range order whose connected correlations decay like1/x in space or 1/t^½ 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 ≈ 35,000; for L < L_c we show numerical evidence for stretched exponential decay of temporal correlations with an exponent consistent with the EW value β_ew = ¼ .