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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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PACS. 42.65.Sf -Dynamics of nonlinear optical systems; optical instabilities, optical chaos and complexity, and optical spatio-temporal dynamics. PACS. 42.60.Mi -Dynamical laser instabilities; noisy laser behavior. PACS. 42.65.Pc -Optical bistability, multistability, and switching.Abstract. -Using a normal form analysis, we derive general conditions under which tetrahedral crystals are stable. This result is applied to two driven nonlinear optical systems: degenerate optical parametric oscillator and passive cavities with defocusing Kerr media, for which we prove that tetrahedral crystals are always unstable close to the Turing instability. This result arises from the interaction of the quasi-neutral mode with the Turing modes. Without this interaction, tetrahedral crystals may be stable.Spontaneous pattern formation is a universal behavior observed in a wide variety of contexts in nonequilibrium systems [1,2]. Such spatial patterns appear when the homogeneous steady state becomes unstable with respect to a Turing bifurcation [3,4]. This instability is caused by a competition between two processes: nonlinearities and transport process like diffusion or/and diffraction. The balance between the two processes generates spontaneously self-organized or ordered patterns. They are characterized by an intrinsic wavelength that is determined by dynamical parameters and not by the geometrical constraints imposed by the boundaries. These patterns arise from order parameter equations with the structure ∂u/∂t = N n (u) + Du, where N n (u) is a polynomial of degree n in u and D is a differential operator [5]. The standard equations correspond to D = aL or D = aL + bL 2 , where a and b are coefficients which are either both real or both imaginary, and L is the Laplace operator in one or more dimension.We consider three-dimensional structures in an isotropic system that exhibits a patternforming instability. In the literature, the classification of the various 3D structures or dissipative crystals has been reported in the theory of solidification and crystal nucleation [6]. Some of them have been reported for reaction-diffusion systems [7]. The role of 3D topological defects has also been studied in extended nonequilibrium systems [8]. In optics the interest of 3D pattern formation in the profile of the electromagnetic fields has started very recently. It was predicted that when diffraction competes with chromatic dispersion in a nonlinear ring cavity, the homogeneous intracavity field can be destabilized by diffraction and dispersion, and allows for the formation of three-dimensional crystals [9,10]. For a coherently driven passive ring cavity with a Kerr medium, taking into account both diffraction and chromatic dispersion c EDP Sciences

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