Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

Kozyreff, Gregory; Tlidi, Mustapha

doi:10.1063/1.2759436

Cited by 57 publications

(48 citation statements)

References 54 publications

(45 reference statements)

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“…for q ̸ = 0 [20]. The nonvariational term qw∇ 2 w ensures that (4) does not have a Lyapunov functional so it has the potential to support periodic solutions such as rotating spiral patterns on the sphere.…”

Section: Spirals In Numerical Simulationsmentioning

confidence: 99%

Symmetric Spiral Patterns on Spheres

Sigrist¹,

Matthews²

2011

SIAM J. Appl. Dyn. Syst.

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Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics degree ℓ and ℓ+1. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each ℓ. For small values of ℓ, the centre manifold equations are constructed and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions, or tertiary bifurcations. The results are consistent with numerical simulations of a model pattern-forming system.

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Section: Spirals In Numerical Simulationsmentioning

confidence: 99%

Symmetric Spiral Patterns on Spheres

Sigrist¹,

Matthews²

2011

SIAM J. Appl. Dyn. Syst.

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“…Moreover, it can be derived in a specific limiting in many contexts [49][50][51] and it is well-known that close to bifurcation points, the dynamics is universal, i.e. that its qualitative features are model-independent [52].…”

Section: Introductionmentioning

confidence: 99%

“…The influence of non-variational effects was recently listed in a series of open questions regarding localized patterns [53] and the Swift-Hohenberg equation can definitely not answer it. Secondly, although this equation and non-variational variants of it can be derived in specific limits [51], its range of validity as an asymptotic reduction of more general models is not known.…”

Section: Introductionmentioning

confidence: 99%

Localized Turing patterns in nonlinear optical cavities

Kozyreff

2012

Physica D: Nonlinear Phenomena

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The subcritical Turing instability is studied in two classes of models for laser-driven nonlinear optical cavities. In the first class of models, the nonlinearity is purely absorptive, with arbitrary intensity-dependent losses. In the second class, the refractive index is real and is an arbitrary function of the intra-cavity intensity. Through a weakly nonlinear analysis, a Ginzburg-Landau equation with quintic nonlinearity is derived. Thus, the Maxwell curve, which marks the existence of localized patterns in parameter space, is determined. In the particular case of the Lugiato-Lefever model, the analysis is continued to seventh order, yielding a refined formula for the Maxwell curve and the theoretical curve is compared with recent numerical simulation by Gomilà, Firth, and Scroggie [Physica D 227, 70 (2007).]

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“…Without the delayed feedback, we recover the Swift-Hohenberg equation (SHE) derived in [66,67]. It is one of the most studied partial differential equation in various areas of nonlinear science [68][69][70]. It constitutes a paradigmatic evolution equation that exhibits periodic spatio-temporal patterns as well as localized structures [6,7,11,71,72].…”

Section: Derivation Of the Swift-hohenberg Equation With Delaymentioning

confidence: 61%

Spontaneous motion of localized structures and localized patterns induced by delayed feedback

et al. 2010

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Abstract.We study the properties of 2D dissipative structures in a coherently driven optical resonator subjected to a delayed feedback. It has been predicted that delayed feedback can lead to the spontaneous motion of bright localized structures [M. Tlidi et al., Phys. Rev. Lett. 103, 103904 (2009)]. We study here the phenomenon in detail. In particular, we show that the delayed feedback induces a spontaneous motion of periodic patterns and dark localized structures. We focus our analysis on nascent optical bistability regime where the space time dynamics is described by a variational Swift-Hohenberg equation. In the absence of delayed feedback, dark localized structures and patterns do not move. This behavior occurs when the product of the delay time and the feedback strength exceeds some critical value.

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Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

Cited by 57 publications

References 54 publications

Symmetric Spiral Patterns on Spheres

Symmetric Spiral Patterns on Spheres

Localized Turing patterns in nonlinear optical cavities

Spontaneous motion of localized structures and localized patterns induced by delayed feedback

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