Stability of patterns with arbitrary period for a Ginzburg-Landau equation with a mean field

Norbury, John W.; Wei, Juncheng; Winter, Matthias

doi:10.1017/s0956792507006894

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…2(a). Equations of this type arise in numerous applications [22,[78][79][80] and their properties have been studied in several papers [79,[81][82][83]. We mention, in particular, that unmodulated wavetrains bifurcate supercritically when ξ 2 < 54, a condition that differs from the corresponding condition ξ 2 < 54/19 for spatially modulated wavetrains.…”

Section: Appendixmentioning

confidence: 97%

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

et al. 2013

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The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.

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Section: Appendixmentioning

confidence: 97%

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

et al. 2013

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Convectons in a rotating fluid layer

et al. 2013

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Two-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time-independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions.

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Localized structures and front propagation in systems with a conservation law

Knobloch

2016

IMA J Appl Math

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Stability of patterns with arbitrary period for a Ginzburg-Landau equation with a mean field

Cited by 3 publications

References 22 publications

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Convectons in a rotating fluid layer

Localized structures and front propagation in systems with a conservation law

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