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.

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.

Quasi-three dimensional analysis of global instabilities: onset of vortex shedding behind a wavy cylinder

2011 ◽  
Vol 677 ◽  
pp. 572-588 ◽  
Author(s):  
A. GARBARUK ◽  
J. D. CROUCH
Keyword(s):  
Stability Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Vortex Shedding ◽  
Eigenvalue Problems ◽  
Landau Equation ◽  
Three Dimensional ◽  
Maximum Diameter ◽  
Ginzburg Landau ◽  
The Stability

In this paper the global-stability theory is extended to account for weak spanwise-flow variations using a quasi-three-dimensional framework. The analysis considers the onset of vortex shedding behind a circular cylinder with a spanwise-varying diameter. The quasi-three-dimensional approach models the fully three-dimensional flow structure as a series of two-dimensional eigenvalue problems representing the sectional-flow behaviour. The sectional results are coupled together using the Ginzburg–Landau equation, which models the diffusive coupling and provides the global response. The onset of global instability (and thus vortex shedding) is linked to both the sectional growth rates (characterized by the maximum-diameter location) and the spanwise extent of the zone of instability. Unsteady numerical simulations are used to guide the global-stability analysis and to assess the fidelity of the predictions. Results from the stability analysis are shown to be in good agreement with the numerical simulations, which are in close agreement with experiments.

Nonlinear Analysis of Rayleigh–Taylor Instability of Cylindrical Flow With Heat and Mass Transfer

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Mukesh Kumar Awasthi
Keyword(s):  
Mass Transfer ◽  
Nonlinear Analysis ◽  
Heat And Mass Transfer ◽  
Landau Equation ◽  
Taylor Instability ◽  
Ginzburg Landau ◽  
Stability Diagrams ◽  
Rayleigh Taylor Instability ◽  
Mass And Heat Transfer ◽  
The Stability

We study the nonlinear Rayleigh–Taylor instability of the interface between two viscous fluids, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. The fluids are considered to be viscous and incompressible with different kinematic viscosities. The method of multiple expansions has been used for the investigation. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It has been observed that the heat and mass transfer has stabilizing effect on the stability of the system in the nonlinear analysis.

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