Mathematical Tools for Pattern Formation

Dangelmayr, Gerhard; Kramer, Lorenz

doi:10.1007/3-540-49537-1_1

Cited by 10 publications

(7 citation statements)

References 102 publications

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“…Numerical simulations of (4.20) in the unstable regimes shown in figure 7 demonstrate a transition to modulated waves for not too large periodicity intervals L, and to modulated waves with 'phase' or 'defect' turbulence for large L. Once the perturbations have grown the dynamics of (4.20) thus seems to be qualitatively the same as in the case of the standard Ginzburg-Landau equation (see e.g. Dangelmayr & Kramer 1998).…”

Section: 4mentioning

confidence: 74%

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Low-Prandtl-number convection in a rotating cylindrical annulus

Plaut

Busse

2002

J. Fluid Mech.

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Motivated by recent experimental results obtained in a low-Prandtl-number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde mean flow develops at quadratic order. The Poiseuille part of this mean flow is determined by an equation obtained by averaging of the Navier–Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a short-wavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.

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Section: 4mentioning

confidence: 74%

“…In order to study the nonlinear dynamics of the system near onset, i.e. for 0 < = R/R c − 1 1, we construct the Ginzburg-Landau or envelope equation with the spectral method as reviewed for instance by Dangelmayr & Kramer (1998).…”

Section: Weakly Nonlinear Dynamics: Envelope Equationmentioning

confidence: 99%

Low-Prandtl-number convection in a rotating cylindrical annulus

Plaut

Busse

2002

J. Fluid Mech.

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“…The cubic complex Ginzburg-Landau equation (CGLe) is one of the most-studied nonlinear equations in the physics community. It describes on a qualitative, and often even on a quantitative level a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystals and strings in field theory (Kuramoto, 1984, Cross and Hohenberg, 1993, Pismen, 1999, Bohr et al, 1998, Dangelmayr and Kramer, 1998.…”

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confidence: 99%

The world of the complex Ginzburg-Landau equation

2002

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The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to secondorder phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystals and strings in field theory. Our goal is to give an overview of various phenomena described the complex Ginzburg-Landau equation in one, two and three dimensions from the point of view of condensed matter physicists. Our approach is to study the relevant solutions to get an insight into nonequilibrium phenomena in spatially extended systems.

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“…All admissible values of β form a manifold B = {β}. In many cases, differential equations in partial derivatives can be reduced to a d-dimensional system of ordinary differential equations, with a dimension d that may equal infinity [2]. For the simplicity of notation, we shall keep in mind this possibility of working with a d-dimensional dynamical system.…”

Section: Probability Distribution Of Patternsmentioning

confidence: 99%

Probabilistic approach to pattern selection

Yukalov

2001

Physica A: Statistical Mechanics and its Applications

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The problem of pattern selection arises when the evolution equations have many solutions, whereas observed patterns constitute a much more restricted set. An approach is advanced for treating the problem of pattern selection by defining the probability distribution of patterns. Then the most probable pattern naturally corresponds to the largest probability weight. This approach provides the ordering principle for the multiplicity of solutions explaining why some of them are more preferable than other. The approach is applied to solving the problem of turbulent photon filamentation in resonant media.

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Mathematical Tools for Pattern Formation

Cited by 10 publications

References 102 publications

Low-Prandtl-number convection in a rotating cylindrical annulus

Low-Prandtl-number convection in a rotating cylindrical annulus

The world of the complex Ginzburg-Landau equation

Probabilistic approach to pattern selection

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