Instability of spatially quasi-periodic states of the Ginzburg-Landau equation

1994 ◽  
Vol 444 (1921) ◽  
pp. 347-362 ◽  
Keyword(s):  
Linear Stability ◽  
Hydrodynamic Stability ◽  
Model Equation ◽  
Landau Equation ◽  
Geometric Interpretation ◽  
Sufficient Condition ◽  
Ginzburg Landau ◽  
Frequency Map ◽  
The Stability ◽  
Two Parameter

The Ginzburg-Landau (GL) equation with real coefficients is a model equation appearing in superconductor physics and near-critical hydrodynamic stability problems. The stationary GL equation has a two-parameter ( I 1 , I 2 ) family of spatially quasi-periodic (QP) states with frequencies ( ω 1 , ω 2 ) and frequency map with determinant ∆ K = ∂( ω 1 , ω 2 ) / ∂( I 1 , I 2 ). In this paper the linear stability of these QP states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of ∆ K : when ∆ K < 0 the spatially QP state is unstable and ∆ K > 0 is a necessary but not sufficient condition for linear stability. There is an interesting relation between ∆ K and the KAM persistence theorem for invariant toroids.

Spatial structure of a nonlinear wave of stratification

1998 ◽  
Vol 60 (2) ◽  
pp. 215-228 ◽  
Author(s):  
M. ROTTMANN ◽  
K. H. SPATSCHEK
Keyword(s):  
Nonlinear Wave ◽  
Model Equation ◽  
Positive Column ◽  
Landau Equation ◽  
Reductive Perturbation Method ◽  
Ginzburg Landau ◽  
Subcritical Regime ◽  
Ginzburg Landau Equation ◽  
Argon Discharge ◽  
Radial Structure

The positive column of a gas discharge is investigated in the subcritical regime when no self-excited nonlinear ionization wave exists. We derive a model equation for the so-called wave of stratification, which is an externally excited envelope wave. The main point of this work is that, in contrast to previous papers, we take into account the radial structure of the homogeneous column. The mathematical description of the wave of stratification in the form of a complex Ginzburg–Landau equation follows by a systematic reductive perturbation method. The coefficients in the Ginzburg–Landau equation are first calculated in general and are then evaluated explicitly for a low-pressure argon discharge. The results are compared with those obtained from a frequently used simpler model that neglects any radial structure. Finally, the dynamics of the nonlinear wave of stratification is demonstrated via numerical simulations.

scholarly journals Bright-Dark and Multi Solitons Solutions of (3 + 1)-Dimensional Cubic-Quintic Complex Ginzburg–Landau Dynamical Equation with Applications and Stability

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 202 ◽  
Author(s):  
Chen Yue ◽  
Dianchen Lu ◽  
Muhammad Arshad ◽  
Naila Nasreen ◽  
Xiaoyong Qian
Keyword(s):  
Dynamical Equation ◽  
Landau Equation ◽  
Applied Physics ◽  
Ginzburg Landau ◽  
Open Set ◽  
Trigonometric Function Solutions ◽  
Kink Waves ◽  
Mathematical Techniques ◽  
The Stability ◽  
Complex Phenomena

In this paper, bright-dark, multi solitons, and other solutions of a (3 + 1)-dimensional cubic-quintic complex Ginzburg–Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburg–Landau equation with broken phase symmetry has strict positive space–time entropy for an open set of parameter values. The exact wave results in the forms of dark-bright solitons, breather-type solitons, multi solitons interaction, kink and anti-kink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model.

Stable Stationary Solitons of the One-Dimensional Modified Complex Ginzburg-Landau Equation

2007 ◽  
Vol 62 (7-8) ◽  
pp. 368-372
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Landau Equation ◽  
Model Parameters ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Paraxial Ray Approximation ◽  
New Family ◽  
Ginzburg Landau Equation ◽  
The Stability ◽  
The One ◽  
Paraxial Ray

We report on the existence of a new family of stable stationary solitons of the one-dimensional modified complex Ginzburg-Landau equation. By applying the paraxial ray approximation, we obtain the relation between the width and the peak amplitude of the stationary soliton in terms of the model parameters. We verify the analytical results by direct numerical simulations and show the stability of the stationary solitons.

Three-dimensional dissipative optical solitons

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Dumitru Mihalache
Keyword(s):  
Landau Equation ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Optical Solitons ◽  
Ginzburg Landau ◽  
Systematic Analysis ◽  
Dissipative Solitons ◽  
Existence Domain ◽  
Existence And Stability ◽  
The Stability

AbstractA brief overview of recent theoretical results in the area of three-dimensional dissipative optical solitons is given. A systematic analysis demonstrates the existence and stability of both fundamental (spinless) and spinning three-dimensional dissipative solitons in both normal and anomalous group-velocity regimes. Direct numerical simulations of the evolution of stationary solitons of the three-dimensional cubic-quintic Ginzburg-Landau equation show full agreement with the predictions based on computation of the instability eigenvalues from the linearized equations for small perturbations. It is shown that the diffusivity in the transverse plane is necessary for the stability of vortex solitons against azimuthal perturbations, while fundamental (zero-vorticity) solitons may be stable in the absence of diffusivity. It has also been found that, at values of the nonlinear gain above the upper border of the soliton existence domain, the three-dimensional dissipative solitons either develop intrinsic pulsations or start to expand in the temporal (longitudinal) direction keeping their structure in the transverse spatial plane.

Time-Dependent Ginzburg-Landau Equation in Car Following Model Considering the Traffic Interruption Probability

2012 ◽  
Vol 198-199 ◽  
pp. 843-847
Author(s):  
Yi Qiang Zhang ◽  
Rong Jun Cheng ◽  
Hong Xia Ge
Keyword(s):  
Landau Equation ◽  
Reductive Perturbation Method ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Car Following ◽  
Second Derivatives ◽  
Spinodal Line ◽  
Car Following Model ◽  
The Stability ◽  
Derivatives Of

This paper focuses on a car-following model which involves the effects of traffic interruption probability. The stability condition of the model is obtained through the linear stability analysis. The time-dependent Ginzburg-Landau (TDGL) equation is derived by the reductive perturbation method. In addition, the coexisting curve and the spinodal line are obtained by the first and second derivatives of the thermodynamic potential. The analytical results show that the traffic interruption probability indeed has an influence on driving behaviour.

A New Algorithm for Testing the Stability of a Polytope: A Geometric Approach for Simplification

1996 ◽  
Vol 118 (3) ◽  
pp. 611-615 ◽  
Author(s):  
Jinsiang Shaw ◽  
Suhada Jayasuriya
Keyword(s):  
Characteristic Equation ◽  
Robust Stability ◽  
Geometric Structure ◽  
Geometric Approach ◽  
Geometric Interpretation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Edge Theorem

Considered in this paper is the robust stability of a class of systems in which a relevant characteristic equation is a family of polynomials F: f(s, q) = a0(q) + a1(q)s + … + an(q)sn with its coefficients ai(q) depending linearly on q unknown-but-bounded parameters, q = (p1, p2, …, pq)T. It is known that a necessary and sufficient condition for determining the stability of such a family of polynomials is that polynomials at all the exposed edges of the polytope of F in the coefficient space be stable (the edge theorem of Bartlett et al., 1988). The geometric structure of such a family of polynomials is investigated and an approach is given, by which the number of edges of the polytope that need to be checked for stability can be reduced considerably. An example is included to illustrate the benefit of this geometric interpretation.

Toroidal equilibrium states with reversed magnetic shear and parallel flow in connection with the formation of Internal Transport Barriers

2015 ◽  
Vol 81 (4) ◽  
Author(s):  
Ap. Kuiroukidis ◽  
G. N. Throumoulopoulos
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Current Density ◽  
Sufficient Condition ◽  
Magnetic Shear ◽  
Transport Barriers ◽  
Internal Transport Barriers ◽  
The Stability ◽  
The Magnetic Field ◽  
Internal Transport

We construct nonlinear toroidal equilibria of fixed diverted boundary shaping with reversed magnetic shear and flows parallel to the magnetic field. The equilibria have hole-like current density and the reversed magnetic shear increases as the equilibrium nonlinearity becomes stronger. Also, application of a sufficient condition for linear stability implies that the stability is improved as the equilibrium nonlinearity correlated to the reversed magnetic shear gets stronger with a weaker stabilizing contribution from the flow. These results indicate synergetic stabilizing effects of reversed magnetic shear, equilibrium nonlinearity and flow in the establishment of Internal Transport Barriers (ITBs).

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