scholarly journals Some stability results for the complex Ginzburg–Landau equation

2019 ◽  
Vol 22 (08) ◽  
pp. 1950038
Author(s):  
Simão Correia ◽  
Mário Figueira
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Bound States ◽  
Landau Equation ◽  
Strong Solutions ◽  
Ginzburg Landau ◽  
Asymptotic Decay ◽  
Ginzburg Landau Equation ◽  
Stability Results

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg–Landau equation (CGL), [Formula: see text] with [Formula: see text], are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when [Formula: see text] and [Formula: see text] large enough.

Weak and strong solutions of the complex Ginzburg-Landau equation

1994 ◽  
Vol 71 (3) ◽  
pp. 285-318 ◽  
Author(s):  
Charles R Doering ◽  
John D Gibbon ◽  
C David Levermore
Keyword(s):  
Landau Equation ◽  
Strong Solutions ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Weak And Strong Solutions

scholarly journals Bound states and interactions of vortex solitons in the discrete Ginzburg-Landau equation

2012 ◽  
Vol 86 (2) ◽  
Author(s):  
C. Mejía-Cortés ◽  
J. M. Soto-Crespo ◽  
Rodrigo A. Vicencio ◽  
Mario I. Molina
Keyword(s):  
Bound States ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Vortex Solitons

Bound states of dark solitons in the quintic Ginzburg-Landau equation

1998 ◽  
Vol 57 (1) ◽  
pp. 1088-1091 ◽  
Author(s):  
V. V. Afanasjev ◽  
P. L. Chu ◽  
B. A. Malomed
Keyword(s):  
Bound States ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Dark Solitons ◽  
Ginzburg Landau Equation

Instability of quasiperiodic solutions of the Ginzburg–Landau equation

1995 ◽  
Vol 125 (3) ◽  
pp. 501-517 ◽  
Author(s):  
A. Doelman ◽  
R. A. Gardner ◽  
C. K. R. T. Jones
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbit ◽  
Eigenvalue Problem ◽  
Systems Approach ◽  
Landau Equation ◽  
Rigorous Proof ◽  
Unstable Periodic Orbit ◽  
Ginzburg Landau ◽  
Topological Methods ◽  
Ginzburg Landau Equation

In this paper we show that each quasiperiodic standing wave solution of the real Ginzburg–Landau equation which is on the global branch emanating from the Eckhaus unstable periodic orbit is itself unstable. A rigorous proof of the instability is given by showing that the linearised operator about such a solution has spectrum which contains an interval along the unstable axis of the spectral plane. The proof employs some geometric and topological methods arising from a dynamical systems approach to the analysis of the eigenvalue problem for the linearised operator.

scholarly journals Transforming Analytically Intractable Dynamical Systems with a Control Parameter into a Tractable Ginzburg-Landau Equation: Few Illustrations

2018 ◽  
Vol 35 (1-2) ◽  
pp. 35-44 ◽  
Author(s):  
C. Kanchana ◽  
P.G. Siddheshwar
Keyword(s):  
Dynamical Systems ◽  
Control Parameter ◽  
Landau Equation ◽  
Chen System ◽  
Third Order ◽  
Ginzburg Landau ◽  
First Order ◽  
Multi Scale ◽  
Ginzburg Landau Equation ◽  
Fifth Order

In the paper a means of making a simplified study of dynamical systems with a control parameter is presented. The intractable, third-order classical Lorenz system, the Lorenz-like Chen system and two topologically dissimilar fifth-order Lorenz systems are considered for illustration. Using the multi-scale method, these systems are reduced to an analytically tractable first-order Ginzburg-Landau equation (GLE) in one of the amplitudes. The analytical solution of the GLE is used to find the remaining amplitudes.

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