Complex Ginzburg-Landau Model Research Articles

Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated [formula omitted]-fractional derivative, beta fractional derivative, and conformable fractional derivative types

The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta-surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M-fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α1=2α2, the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α2=-ρ+6α18, the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations.

Breakdown of the Meissner effect at the zero exceptional point in non-Hermitian two-band BCS model

The spontaneous symmetry breaking of a continuous symmetry in complex field theory at the exceptional point (EP) of the parameter space is known to exhibit interesting phenomena, such as the breakdown of a Higgs mechanism. In this work, we derive the complex Ginzburg–Landau model from a non-Hermitian two-band Bardeen–Cooper–Schrieffer model via path integral and investigate its spontaneous symmetry breaking. We find that analog to the Higgs mechanism, the Meissner effect of the complex Ginzburg–Landau model also breaks down at the EP while the gap parameters stay finite.

Investigating the noise effect on the CGL model having parabolic law of nonlinearity

The stochastic complex Ginzburg–Landau model with parabolic law nonlinearity and chromatic dispersion is investigated in this research study. In many fields of science, including fluid dynamics and optical fibers, this model is utilized to represent the wave nature. In this study, the improved modified extended tanh approach is used. This method provides a wide variety of solutions, including dark, singular and bright solitons. Furthermore, exponential, singular periodic and Weierstrass elliptic solutions are provided. For a variety of nonlinear partial differential equations, this method offers a practical and effective method for determining exact solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths.

A study of variation in dynamical behavior of fractional complex Ginzburg-Landau model for different fractional operators

A study of variation in dynamical behavior of fractional complex Ginzburg-Landau model for different fractional operators

Dynamics of the (2+1)$$ \left(2+1\right) $$‐dimensional coupled cubic–quintic complex Ginzburg–Landau model for convecting binary fluid: Analytical investigations and multiple‐soliton solutions

The ‐dimensional coupled cubic–quintic complex Ginzburg–Landau equations ( ‐DCC‐QCGLEs) can simulate a variety of binary fluid thermal convection characteristics, containing complex parameters. The analysis of pattern formation in chaotic and nonlinear dynamical systems can benefit greatly from the use of this thermal convection model. The primary goal of this study is to find analytical solutions for some recent advances that have been made for Rayleigh–Bénard convection by applying the ‐DCC‐QCGLEs model for slowly varying spatio‐temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. Furthermore, we introduce the WTC‐Kruskal algorithm of the Painlevé methodology to examine the integrability of the ‐DCC‐QCGLEs and the truncated Painlevé expansion is used to extract the Bäcklund transform, from which new solitary solutions can be acquired. The results also demonstrated a good agreement with previous works and were more significant and accurate in two and three dimensions of the proposed model. Finally, the computational results indicate that the effects of the physical parameters of the considered equations can be demonstrated by utilizing 2D and 3D graphics for different values of these parameters.

Analytical Solutions of the Fractional Complex Ginzburg-Landau Model Using Generalized Exponential Rational Function Method with Two Different Nonlinearities

The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order α . Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model.

Optical dromions for complex Ginzburg Landau model with nonlinear media

Optical dromions for complex Ginzburg Landau model with nonlinear media

A study of fractional complex Ginzburg–Landau model with three kinds of fractional operators

The fractional complex Ginzburg–Landau equation plays an important role in the field of optics, field theory and superconductivity. In this paper, two variable G′G,1G–expansion method has been used on the fractional complex Ginzburg–Landau equation. The governing model is studied with three kinds of nonlinearities, namely; Kerr, quadratic–cubic and parabolic laws. Each law applied on fractional complex Ginzburg–Landau model is further discussed along with three definitions of the derivative i.e., beta derivative, conformable derivative and M-truncated derivative. In order to observe the fractional effects, a graphical comparison is shown. The evolution of solutions of the governing model for increasing value of fractional parameter is depicted through the surface graphs and line plots.

The investigation of several soliton solutions to the complex Ginzburg-Landau model with Kerr law nonlinearity

This work investigates the complex Ginzburg-Landau equation (CGLE) with Kerr law in nonlinear optics, which represents soliton propagation in the presence of a detuning factor. The φ^6-model expansion approach is used to find optical solitons such as dark, bright, singular, and periodic as well as the combined soliton solutions to the model. The results presented in this study are intended to improve the CGLE's nonlinear dynamical characteristics, it might also assist in comprehending some of the physical implications of various nonlinear physics models. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cotangent appears in the Langevin function for magnetic polarization. The current research is frequently used to report a variety of fascinating physical phenomena, such as the Kerr law of non-linearity, which results from the fact that an external electric field causes non-harmonic motion of electrons bound in molecules, which causes nonlinear responses in a light wave in an optical fiber. The obtained solutions' 2-dimensional, 3-dimensional, and contour plots are shown.

Gravitational Modulation Effect on Double-Diffusive Oscillatory Convection in a Viscoelastic Fluid Layer

The complex Ginzburg-Landau model has been employed to derive the temporal amplitude of convection in a viscoelastic fluid layer under gravity modulation. The finite amplitude is based on a weakly nonlinear thermal convection. The perturbation technique is applied to simplify the nonlinear system of partial differential equations into a non-autonomous amplitude equation. Heat and mass transfer obtained in terms of the Nusselt and Sherwood numbers, and the corresponding results are presented for the small amplitude of modulation. The moderate values of amplitude and frequencies of gravity modulation are considered, their effect is to enhance the heat and mass transfer in the layer. Thus, the modulation plays dual role in heat and mass transfer. The variations of Nusselt and Sherwood numbers with slow time becomes rapid on either increasing the parameters Rs, Pr, λ, δ or decreasing the parameters Γ, ε, Ω. Each parameter effect has been presented graphically with their effect on Nu and Sh. Further, it is found that an oscillatory mode of convection enhances the heat and mass transfer than a stationary mode.

Conservation laws for pure-cubic optical solitons with complex Ginzburg–Landau equation having several refractive index structures

This paper derives the conserved densities for the perturbed complex Ginzburg–Landau model which is addressed with a range of nonlinear forms. The densities are derived with the implementation of Lie symmetry analysis while the conserved quantities are obtained from the soliton solutions to the model. For two such nonlinear forms the Hamiltonian cease to exist since the corresponding integrals are rendered divergent.

Simple harmonic and damped motions of dissipative solitons in two-dimensional complex Ginzburg-Landau equation supported by an external V-shaped potential

Dissipative solitons based on the complex Ginzburg-Landau (CGL) model show many novel dynamic properties. In this paper, a series of novel simple harmonic and damped motion dynamics of soliton supported by induced V-shaped potential in the cubic-quintic CGL model was investigated. Without viscosity, the role of these potential wells in stabilizing dissipative soliton forms periodic oscillation, just like simple harmonic motion. The influence of potential slope and oscillating amplitude on the period and momentum of simple harmonic motion were numerically analyzed. By adding a small diffusivity term (viscosity) into the CGL model, a significant damping effect is applied to the simple harmonic motion of dissipative solitons. The evolution mechanism of the energy and momentum during the simple harmonic motion and the damped motion was numerically studied. In addition, the energy gain/loss in the CGL model has no impact on the dynamical evolution of simple harmonic motion and damped motion of dissipative solitons.

The generalized complex Ginzburg–Landau model and its dark and bright soliton solutions

The generalized complex Ginzburg–Landau model and its dark and bright soliton solutions

New complex wave structures to the complex Ginzburg-Landau model

<abstract> Int his paper, we study and analysis the complex Ginzburg-Landau model or CGL model to obtain some new solitary wave structures through the modified $ (G'/G) $-expansion method. Those solutions can explain through hyperbolic, trigonometric, and rational functions. The graphical design makes the dynamics of the equations noticeable. Herein, we state that the examined method is important, powerful, and significant in performing numerous solitary wave structures of various nonlinear wave models following in physics and engineering as well. </abstract>

Optical solitons for complex Ginzburg–Landau model with Kerr, quadratic–cubic and parabolic law nonlinearities in nonlinear optics by the $$\hbox {exp}(-\Phi (\zeta ))$$ expansion method

The optical solitons for the complex Ginzburg–Landau model with Kerr law, quadratic–cubic law and parabolic law are obtained via the $$\hbox {exp}(-\Phi (\zeta ))$$ expansion method. Many abundant solutions such as complex dark-singular, complex periodic-singular and plane-wave solutions are derived for this model. These complex solutions are useful for understanding the physical properties for this model. Figures are presented for these solutions to show the dynamics for these waves.

Fractal modification of complex Ginzburg–Landau model arising in the oscillating phenomena

The complex Ginzburg-Landau Equation (CGLE) is one of the non-trivial models for addressing the dynamics of oscillating, highly nonlinear processes right before the start of oscillations. This paper presents the complex Ginzburg-Landau fractal model with three types of nonlinearity. The variational approach provides soliton solutions for the CGLE in terms of Kerr, parabolic, and quadratic laws of nonlinearity. New bright and dark soliton solutions for the CGLE are developed. The necessary novel conditions that guarantee the existence of suitable solitary waves are introduced. Monitoring solutions 3D and 2D plots are illustrated by choosing a range of appropriate values of parameters.

Continuous-wave stability and multipulse structures in a universal complex Ginzburg–Landau model for passively mode-locked lasers with a saturable absorber

The stability of continuous waves and the dynamics of multipulse structures are studied theoretically for a generic model of passively mode-locked fiber lasers with arbitrary nonlinearity. The model is characterized by a complex Ginzburg–Landau equation with a nonlinear term I m / ( 1 + Γ I ) n , where I is the field intensity, m and n are real numbers, and Γ is the saturation power. Fixed-point solutions of the governing equations reveal an interesting loci of singular points in the amplitude-frequency plane consisting of zero, one, or two fixed points, depending on the values of m and n . The continuous-wave stability is analyzed within the framework of the modulational-instability theory, and the results demonstrate a bifurcation in the continuous-wave amplitude growth rate and propagation constant characteristic of multiperiodic wave structures. In the full nonlinear regime, these multiperiodic wave structures turn out to be multipulse trains, unveiled via numerical simulations of the model nonlinear equation, the rich variety of which is highlighted by considering different combinations of values for the pair ( m , n ). The results are consistent with previous analyses of the dynamics of multipulse structures in several contexts of passively mode-locked lasers with a saturable absorber, as well as with predictions about the existence of multipulse structures and bound-state solitons in optical fibers with strong optical nonlinearity, such as cubic-quintic and saturable nonlinearities.

Observation of gain-pinned dissipative solitons in a microcavity laser

We demonstrate an experimental approach to create dissipative solitons in a microcavity laser. In particular, we shape the spatial gain profile of a quasi-one-dimensional microcavity laser with a nonresonant, pulsed optical pump to create spatially localised structures, called gain-pinned dissipative solitons that exist due to the balance of gain and nonlinear losses and are confined to a diffraction-limited volume. The ultrafast formation dynamics and decay of the gain-pinned solitons are probed directly, showing that they are created on a picosecond timescale, orders of magnitude faster than laser cavity solitons. All of the experimentally observed features and dynamics are reconstructed by using a standard complex Ginzburg-Landau model.

The Complex Ginzburg Landau Model for an Oscillatory Convection in a Rotating Fluid Layer

A weakly nonlinear thermal instability is investigated under rotation speed modulation. Using the perturbation analysis, a nonlinear physical model is simplified to determine the convective amplitude for oscillatory mode. A non-autonomous complex Ginzburg-Landau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The effect of rotation is found either to stabilize or destabilize the system. The Nusselt number is obtained numerically to present the results of heat transfer. It is found that modulation has a significant effect on heat transport for lower values of ω&lt;i&gt;f&lt;/i&gt; while no effect for higher values. It is also found that modulation can be used alternately to control the heat transfer in the system. Further, oscillatory mode enhances heat transfer rather than stationary mode.

Complex wave structures for abundant solutions related to the complex Ginzburg–Landau model

In this paper, we apply an analytical algorithm, namely the modified auxiliary equation method, to investigate the complex wave structures for abundant solutions related to the complex Ginzburg–Landau model. That is for studying more physical properties of this model. This model describes the wave profile showing in different physical systems. Many various formulas of solutions are obtained such as complex solitary, soliton, and exponential wave solutions. Some of the solutions are discussed by sketching them in three and two dimensional to show the dynamic of these waves. All obtained solutions are verified of its validity by using Maple software program.