scholarly journals FRONT STRUCTURES IN A REAL GINZBURG-LANDAU EQUATION COUPLED TO A MEAN FIELD

1994 ◽  
Vol 04 (05) ◽  
pp. 1343-1346 ◽  
Author(s):  
HENAR HERRERO ◽  
HERMANN RIECKE
Keyword(s):  
Analytical Study ◽  
Landau Equation ◽  
Mean Field ◽  
Travelling Wave ◽  
Opposite Orientation ◽  
Ginzburg Landau ◽  
Separation Ratio ◽  
Ginzburg Landau Equation ◽  
Front Solutions ◽  
Wave Trains

Localized travelling wave trains or pulses have been observed in various experiments in binary mixture convection. For strongly negative separation ratio, these pulse structures can be described as two interacting fronts of opposite orientation. An analytical study of the front solutions in a real Ginzburg-Landau equation coupled to a mean field is presented here as a first approach to the pulse solution. The additional mean field becomes important when the mass diffusion in the mixture is small as is the case in liquids.

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

2007 ◽  
Vol 18 (2) ◽  
pp. 129-151 ◽  
Author(s):  
J. NORBURY ◽  
J. WEI ◽  
M. WINTER
Keyword(s):  
Eigenvalue Problems ◽  
Landau Equation ◽  
Mean Field ◽  
Complete Classification ◽  
Steady States ◽  
Spatial Average ◽  
Ginzburg Landau ◽  
Nonlinear Functional ◽  
Ginzburg Landau Equation ◽  
Image Position

We consider the following system of equations: where the spatial average ⟨ B ⟩ = 0 and μ > σ > 0. This system plays an important role as a Ginzburg-Landau equation with a mean field in several areas of the applied sciences and the steady-states of this system extend to periodic steady-states with period L on the real line which are observed in experiments. Our approach is by combining methods of nonlinear functional analysis such as nonlocal eigenvalue problems and the variational characterization of eigenvalues with Jacobi elliptic integrals. This enables us to give a complete classification of all stable steady-states for any positive L.

Nonlinear dynamics near the stability margin in rotating pipe flow

1991 ◽  
Vol 233 ◽  
pp. 329-347 ◽  
Author(s):  
Z. Yang ◽  
S. Leibovich
Keyword(s):  
Pipe Flow ◽  
Landau Equation ◽  
Stability Boundary ◽  
Travelling Wave ◽  
Unstable Wave ◽  
Marginal Stability ◽  
Wave System ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Rotating Pipe Flow

The nonlinear evolution of marginally unstable wave packets in rotating pipe flow is studied. These flows depend on two control parameters, which may be taken to be the axial Reynolds number R and a Rossby number, q. Marginal stability is realized on a curve in the (R, q)-plane, and we explore the entire marginal stability boundary. As the flow passes through any point on the marginal stability curve, it undergoes a supercritical Hopf bifurcation and the steady base flow is replaced by a travelling wave. The envelope of the wave system is governed by a complex Ginzburg–Landau equation. The Ginzburg–Landau equation admits Stokes waves, which correspond to standing modulations of the linear travelling wavetrain, as well as travelling wave modulations of the linear wavetrain. Bands of wavenumbers are identified in which the nonlinear modulated waves are subject to a sideband instability.

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