The Geometry of the Phase Diffusion Equation

We study the existence of interfaces between stripe or roll solutions in the Swift-Hohenberg equation. We prove the existence of two different types of interfaces: corner-like interfaces, also referred to as knee solutions, and step-like interfaces. The analysis relies upon a spatial dynamics formulation of the existence problem and an equivariant center manifold reduction. In this setting, the interfaces are found as heteroclinic and homoclinic orbits of a reduced system of ODEs.

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“…We refer to [7,8] and the references therein for a discussion of these Our main results concern the existence of these two types of solutions.…”

Section: Introductionmentioning

confidence: 99%

Interfaces between rolls in the Swift-Hohenberg equation

Hărăguş

Scheel²

2007

IJDSDE

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“…The energy associated with the variational problem (3.1) has the form of the well-known GinzburgLandau functional [1]. The difference is that in the RCN variational problem this energy is restricted to vector fields which are gradient.…”

Section: Description Of Asymptotic Minimizers For Rcnmentioning

confidence: 99%

“…We recently established [1] rigorous proofs for the existence and form of solutions to the regularized phase diffusion equation in the limit where the inverse aspect ratio ε (ratio of the roll wavelength to the typical size of the container) goes to zero. The basis of this work was first to remark that solving the phase diffusion equation amounts to solving a mini-mization problem for an energy E which is the sum of two terms, one associated with roll bending and the other with the mismatch between the local and the preferred wave number.…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 briefly describes the main results of our previous work [1]. In Section 4, we introduce the self-dual equation and the exact phase grain boundary (PGB) solution.…”

Section: Introductionmentioning

confidence: 99%

“…(2.3) to describe the resulting dynamics. Moreover, the stationary phase diffusion equation admits multivalued solutions [1] which can only relate to the physically relevant solutions if one adds a regularization coming from the contributions of the next orders of the asymptotic expansion. The dominant correction comes at order ε 3 and can be approximated by a linear term of the form ε 2 η∇ 4 Θ.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global description of patterns far from onset: a case study

Ercolani

Indik

Newell

et al. 2003

Physica D: Nonlinear Phenomena

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The Cross-Newell phase diffusion equation τ(k)Θ T = −∇ · kB(k), k = ∇Θ, | k| = k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-trivial case study of high Prandtl number convection in a large aspect ratio, shallow, elliptical container with heated sidewalls, for which we also have the results of simulation and experiment.

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Phase dynamics in the real Ginzburg‐Landau equation

Melbourne

Schneider

2003

Mathematische Nachrichten

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Spatially periodic equilibria A(X, T) = √1 − q2 e italiciqX+italiciϕ 0 are the locally preferred planform for the Ginzburg‐Landau equation ∂TA = ∂2XA + A − A|A|2. To describe the global spatial behavior, an evolution equation for the local wave number q can be derived formally. The local wave number q satisfies approximately a so called phase diffusion equation ∂τq = ∂2ξh(q). It is the purpose of this paper to explain the extent to which the phase diffusion equation is valid by proving estimates for this formal approximation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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The Geometry of the Phase Diffusion Equation

Cited by 47 publications

References 48 publications

Interfaces between rolls in the Swift-Hohenberg equation

Interfaces between rolls in the Swift-Hohenberg equation

Global description of patterns far from onset: a case study

Phase dynamics in the real Ginzburg‐Landau equation

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