scholarly journals Periodic waves of the Lugiato–Lefever equation at the onset of Turing instability

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170188 ◽  
Author(s):  
Lucie Delcey ◽  
Mariana Haraguss
Keyword(s):  
Orbital Stability ◽  
Stability Result ◽  
Turing Instability ◽  
Theme Issue ◽  
Centre Manifold ◽  
Periodic Perturbations ◽  
Centre Manifold Reduction ◽  
The Stability ◽  
Bounded Perturbations ◽  
Manifold Reduction

We study the existence and the stability of periodic steady waves for a nonlinear model, the Lugiato–Lefever equation, arising in optics. Starting from a detailed description of the stability properties of constant solutions, we then focus on the periodic steady waves which bifurcate at the onset of Turing instability. Using a centre manifold reduction, we analyse these Turing bifurcations, and prove the existence of periodic steady waves. This approach also allows us to conclude on the nonlinear orbital stability of these waves for co-periodic perturbations, i.e. for periodic perturbations which have the same period as the wave. This stability result is completed by a spectral stability result for general bounded perturbations. In particular, this spectral analysis shows that instabilities are always due to co-periodic perturbations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

scholarly journals Pattern formation in a slowly flattening spherical cap: delayed bifurcation

2020 ◽  
Vol 85 (4) ◽  
pp. 513-541
Author(s):  
Laurent Charette ◽  
Colin B Macdonald ◽  
Wayne Nagata
Keyword(s):  
Normal Form ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Centre Manifold ◽  
Spherical Cap ◽  
Centre Manifold Reduction ◽  
Delayed Bifurcation ◽  
The Stability ◽  
Parameter Values ◽  
Manifold Reduction

Abstract This article describes a reduction of a non-autonomous Brusselator reaction–diffusion system of partial differential equations on a spherical cap with time-dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this non-autonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions.

scholarly journals Grain boundaries in the Swift–Hohenberg equation

2012 ◽  
Vol 23 (6) ◽  
pp. 737-759 ◽  
Author(s):  
MARIANA HARAGUS ◽  
ARND SCHEEL
Keyword(s):  
Grain Boundaries ◽  
Order Approximation ◽  
Spatial Dynamics ◽  
Heteroclinic Orbits ◽  
Existence Problem ◽  
Leading Order ◽  
Centre Manifold ◽  
Centre Manifold Reduction ◽  
Manifold Reduction ◽  
Wavenumber Selection

We study the existence of grain boundaries in the Swift–Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ordinary differential equations in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

Combustion wave speed

1995 ◽  
Vol 450 (1938) ◽  
pp. 193-198 ◽  
Keyword(s):  
Combustion Wave ◽  
Wave Speed ◽  
Numerical Calculations ◽  
Reaction Function ◽  
Centre Manifold ◽  
Centre Manifold Reduction ◽  
Manifold Reduction

In this paper, we determine the ignition wave speed and the extinction wave speed for a reaction function which was obtained by centre-manifold reduction and is relevant to combustion. Our numerical calculations cover the entire relevant δ range for three choices of ∊.

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