PHYSICAL REVIEW E VOLUME 47, NUMBER 6 JUNE 1993 Counterpropagating periodic pulses in coupled Ginzburg-Landau equations Boris A. Malorned* Department of Physics, University of Illinois at Urbana Cha-mpaign, 1110 West Green Street, Urbana, Illinois 61801 (Received 25 February 1993) A recently observed stable regime in the form of periodically colliding counterpropagating wave pack- ets (pulses) in an annular convection channel at very small positive overcriticalities is described analyti- cally in terms of coupled Ginzburg-Landau equations. First, the existence of this regime is demonstrated in the framework of the simplest system including only the group-velocity difference, weak gain, and nonlinear dissipative coupling between two modes. In this approximation, the shape of the counterpro- pagating waves remains indefinite. It is demonstrated that additional dispersive terms, regarded as a small perturbation, provide shaping of the wave packets and also give rise to the deviation of the phase velocity from that for purely linear waves. PACS number(s): 47.27. — i 03.40.Kf In recent experiments [1], Kolodner observed a stable regime of counterpropagation of two pulses (called wave packets in Ref. [1]) of convection in a narrow annular channel filled with a binary liquid heated from below. The experiments were conducted at positive but very small values of the overcriticality e, e =0.000 18— 0. 00166 (at larger e, the so-called dispersive chaos [2] sets in). When far from each other, the wave packets un- dergo a slow growth due to the small overcriticality, which is compensated by losses during their collisions, so that a stable dynamical regime is observed with the mean velocity of the packets close to the group velocity of linear waves at the point @=0. In Ref. [1], it was suggested that the regime observed could be directly interpreted in terms of the system of two coupled Ginzburg-Landau (GL) equations with com- plex coefficients governing envelopes of right- and left- propagating waves. For the first time, the system of cou- pled GL equations for counterpropagating waves was considered in Ref. [3] (however, in Ref. [3] only purely real coefficients were considered). The objective of the present work is to do this in the framework of the sim- plest systems of that type. Usually, the GL equations with complex coefficients can be treated analytically in two cases: (i) when real parts of the coefficients are small in comparison to their imaginary parts, so that each equation is close to the non- linear Schrodinger (NS) equation [4], or (ii) when the imaginary parts are small [5]. Recently, the interaction of two counterpropagating solitonlike pulses was con- sidered in the near-NS regime, assuming weak coupling between the two equations [6]. In that work, a threshold (maximum) value of the relative group velocity admitting fusion of the colliding pulses into a bound state was found. Within the framework of the same approxima- tion, it is, as a matter of fact, trivial to describe a regime similar to that reported in Ref. [1] (this would require one to assume that changes of the solitons' amplitude pro- duced by a collision and by the slow growth between col- lisions are small enough). However, such a description does not seem relevant. It was emphasized in Ref. [1] u, — u„=au — fu f'u, (lb) where the group velocities are chosen to be +1. In spite of the simplicity of Eqs. (1), analytical solutions are not available in an exact form; nevertheless, a certain class of approximate solutions will be found below. It is convenient to introduce new independent variables t+x, r= t — —x— and new unknown quantities U= fuf', V— = fu f'. Then Eqs. (1) reduce to U~ =eU — UV, V =@V — UV . (3) (4a) (4b) Due to the symmetry between the right- and left- traveling waves, one should look for solutions satisfying the identity U(g, r)=— V(r, g) . In the case e =0, a general solution to Eqs. (4) and (5) can be readily found: U, (g, r) =g'(~) [g (r)+g(g) ] Vo(g, ~) =g'(g) [g(~)+g(g) ] (6a) (6b) where g is an arbitrary function of one variable. Accord- ing to Eqs. (3), only positive solutions for U and V are meaningful. Then it is necessary to select solutions periodic in x, to be able to model wave propagation in the that the pulses (wave packets) observed in the experiment looked quasilinear, thus being very different from true solitons. The simplest dynamical model of the counterpropagat- ing waves must include the group-velocity difference, linear gain, and nonlinear cross damping: u, +u =eu — Iul u 1063-651X/93/47(6)/3841(3)/$06. 00 47 R3841 1993 The American Physical Society