scholarly journals Modulation Equation for the Stochastic Swift–Hohenberg Equation with Cubic and Quintic Nonlinearities on the Real Line

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1217
Author(s):  
Wael W. Mohammed
Keyword(s):  
Unbounded Domain ◽  
Real Line ◽  
Additive Noise ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Modulation Equation ◽  
The Real ◽  
Quintic Nonlinearity ◽  
Ginzburg Landau Equation ◽  
Dominant Pattern

The purpose of this paper is to rigorously derive the cubic–quintic Ginzburg–Landau equation as a modulation equation for the stochastic Swift–Hohenberg equation with cubic–quintic nonlinearity on an unbounded domain near a change of stability, where a band of dominant pattern is changing stability. Also, we show the influence of degenerate additive noise on the stabilization of the modulation equation.

scholarly journals Phase dynamics in the real Ginzburg-Landau equation

2004 ◽  
Vol 263-264 (1) ◽  
pp. 171-180 ◽  
Author(s):  
Ian Melbourne ◽  
Guido Schneider
Keyword(s):  
Landau Equation ◽  
Ginzburg Landau ◽  
The Real ◽  
Phase Dynamics ◽  
Ginzburg Landau Equation

scholarly journals Random Attractors for Stochastic Ginzburg-Landau Equation on Unbounded Domains

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Qiuying Lu ◽  
Guifeng Deng ◽  
Weipeng Zhang
Keyword(s):  
Additive Noise ◽  
Dimensional Space ◽  
Landau Equation ◽  
Unbounded Domains ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Ginzburg Landau ◽  
Uniform Estimates ◽  
Ginzburg Landau Equation ◽  
Random Attractors

We prove the existence of a pullback attractor inL2(ℝn)for the stochastic Ginzburg-Landau equation with additive noise on the entiren-dimensional spaceℝn. We show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. We demonstrate that the system possesses a uniqueD-random attractor, for which the asymptotic compactness is established by the method of uniform estimates on the tails of its solutions.

PERIODIC NUCLEATION SOLUTIONS OF THE REAL GINZBURG–LANDAU EQUATION IN A FINITE BOX

2002 ◽  
Vol 12 (10) ◽  
pp. 2219-2228 ◽  
Author(s):  
M. ARGENTINA ◽  
O. DESCALZI ◽  
E. TIRAPEGUI
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Landau Equation ◽  
Plane Waves ◽  
Periodic Boundary Conditions ◽  
Stationary Solutions ◽  
Stable Plane ◽  
Ginzburg Landau ◽  
The Real ◽  
Ginzburg Landau Equation

We study the stationary solutions of the real Ginzburg–Landau equation with periodic boundary conditions in a finite box. We show explicitly how to construct nucleation solutions allowing transitions between stable plane waves.

scholarly journals Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation

2015 ◽  
Vol 13 (04) ◽  
pp. 395-411 ◽  
Author(s):  
Jungho Park ◽  
Philip Strzelecki
Keyword(s):  
Traveling Waves ◽  
Landau Equation ◽  
Basic Solution ◽  
Ginzburg Landau ◽  
Center Manifold Reduction ◽  
Modulation Equation ◽  
Ginzburg Landau Equation ◽  
The One ◽  
Manifold Reduction ◽  
Bifurcation And Stability

We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves.

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