Resonant Patterns through Coupling with a Zero Mode

Abstract

The interaction between a diffusive instability and a quasineutral zero mode favors the onset of resonant structures in systems exhibiting inversion symmetry and stabilizes reentrant hexagonal patterns when this symmetry is absent. PACS numbers: 47.20.Ky, 03.40.Gc, 05.70.Ln In driven systems, structures of hexagonal symmetry are ubiquituous. They have been observed in fields as diverse as hydrodynamics (Benard-Marangoni [1] and nonBoussinesq Rayleigh-Benard [2] convection), chemistry (Turing patterns) [3], and nonlinear optics (tranverse patterns) [4]. All these systems are described by a set of nonlinear partial differential equations for the state vector U BU = N(U, r;V),where N is a vector function of U and its spatial derivatives that describes the various kinetic processes taking place in the system. It also depends on some control parameter r.Generically, in 2D, resonant hexagonal patterns stabilized by triplet interactions appear as the result of a symmetry breaking instability in systems lacking inversion symmetry, i.e. , for which the equality N(U, r; V) = -N( -U, r; V)(2) is violated. In 3D, this property is also responsible for the onset of resonant patterns such as body centered cubic structures (bcc) or hexagonally packed cylinders (hpc) [5]. Some of these 3D structures have recently been obtained in open spatial chemical reactors [6]. In this Letter we study the influence of a quasineutral zero mode generated by a secondary steady-state bifurcation (that may mimic a phase transition) on such symmetry breaking instabilities. We show that it induces resonant structures when inversion symmetry is present.In the converse case it leads to reentrant resonant patterns.We first consider a system described by Eq. (1) and for which the condition Eq. (2) is satisfied. We suppose, to fix the ideas, that for homogeneous conditions, the thermodynamic branch Uo, such that N(Uo, r) = 0, undergoes a "pitchfork bifurcation" at r = r~giving rise to two new homogeneous steady states (HSS): U+ = -U . We furthermore impose that the trivial state Up can be destabilized by inhomogeneous perturbations of wave number q, leading to a diffusive-type instability [3] occurring at r = rT~rp. Standard bifurcation theory may be applied to describe the patterned solutions that branch off the state Up at r = rr [7]. In large aspect ratio systems, the field U may be approximated by any linear superposition of I critical modes BA; m 1 = pAi -gilA I'A -g2 g IA, I'A; (j «), (4) j=1 where p, = (r -rr)/rr and g2 ) gi~0 . Owing to the absence of quadratic terms [Eq. (2)] and because the cubic interaction is competitive (i.e., the cross-mode coupling gq is larger than the self-coupling feedback gi), the only stable structure corresponds to stripes with m = 1 [8]. They. appear supercritically at r = rT with an amplitude Ai such that IAil = Ri = Qp/gi. However, when r r~, the growth rate~p of a homogeneous perturbation of amplitude Ap about the state Up tends to zero. As a result the uniform mode Ap progressively rejoins the set of active mode...

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