Complex Ginzburg-Landau Equation Research Articles

Dispersive perturbations of solitons for conformable fractional complex Ginzburg–Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method

Dispersive perturbations of solitons for conformable fractional complex Ginzburg–Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method

Revisitation of “Implicit Quiescent Optical Solitons with Complex Ginzburg-Landau Equation Having Nonlinear Chromatic Dispersion“: Linear Temporal Evolution

The current paper revisits the retrieval of quiescent optical solitons for the complex Ginzburg-Landau equation with nonlinear chromatic dispersion and several forms of self-phase modulation. The results obtained in this work consist of explicit or implicit quiescent optical solitons, unlike the previous report, which were expressed in terms of quadratures. Additionally, this study addresses two additional forms of self-phase modulation: the saturating law and the exponential law. This exploration yields quiescent optical solitons expressed in terms of quadratures for the first time.

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

Soliton-like nonlinear excitation in the FitzHugh–Nagumo cardiac model through the cubic–quintic complex Ginzburg–Landau equation

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Dynamics of femtosecond lasers and induced plasma in non-Kerr nonlinear transparent materials: Competing effects of electron–hole radiative recombinations and single-electron diffusions

Modern laser micromachining utilizes ultrashort optical fields, such as femtosecond lasers, to perform high-precision processings on solid materials, including cutting, drilling, ablation, polishing, and scripturing. Femtosecond laser-based material processings can induce a plasma of free electrons whose density depends on physical phenomena such as single-electron diffusions, multiphoton ionization, and electron–hole radiative recombinations. In this work, we examine the dynamics of femtosecond lasers in transparent materials with non-Kerr nonlinearity, taking into account the generation of an electron plasma. In these specific materials, a balance between the nonlinearity and the group-velocity dispersion of the optical medium can favor the formation of optical filaments propagating with a permanent shape by virtue of their “solitonic” features. We are interested in the effects of the competition between electron–hole radiative recombination and single-electron diffusion processes on the spatiotemporal profiles of the propagating optical field and of the plasma density. The model features a complex Ginzburg–Landau equation with an optical nonlinearity of a general saturable form and a Kth-order nonlinearity term accounting for K-photon ionization processes, coupled to a rate equation for the electron plasma density where the present terms are representing avalanche ionizations, single-electron diffusion, and electron–hole radiative recombination processes. The modulational-instability analysis suggests that the continuous-wave regime will be stabilized by strong electron–hole radiative recombination processes for a fixed value of the single-electron diffusion coefficient, a stability enhanced by an increase in K. In the nonlinear regime, numerical simulations of the model equations for different combinations of the nonlinearity-saturation exponents and different values of the photon number K unveil soliton train structures forming from the laser field propagation and the time evolution of the plasma density. These structures turn out to be either dissipative soliton trains in the absence of electron–hole radiative recombinations or soliton crystals when electron–hole radiative recombination processes are taken into consideration to balance the damping effect caused by single-electron diffusions.

Optical soliton solutions of complex Ginzburg–Landau equation with triple power law and modulation instability

This paper examines the complex Ginzburg Landau equation, which describes pulse propagation inside a fiber with the triple power law of self-phase modulation. Since the effect of parameter selection has become very important in relevant model studies recently, self-phase modulation has been added to the complex Ginzburg Landau equation, which has been studied in the literature, and it is aimed at investigating the analytical solutions of the presented equation. Adding the triple power law of the self-phase modulation parameter to the model, in addition to existing studies in the literature, emphasizes the innovative aspect and importance of the study. The first aim is to reveal bright and singular solitons using the new Kudryashov method. The new Kudryashov method is a technique that is frequently used in the literature, is effective for generating analytical solutions, provides ease of operation, and can be applied to a wide class of nonlinear partial differential equations. The second goal is to show that the obtained solutions have modulation stability. By using modulation instability analysis, the gain spectrum is formed for different parameter values. Graphic presentations support the findings. Moreover, bright and singular soliton portraits are demonstrated with 3D and 2D graphs. The novelty of the study lies in the fact that the relevant model has not been studied before with an effective method such as the new Kudryashov method, and the modulation instability has been studied for the first time.

Chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity

Abstract What the motivation of this paper is to provide chirped optical solitons for the complex Ginzburg–Landau equation with Hamiltonian perturbations and Kerr law nonlinearity. We get 19 exact chirped solutions by utilizing trial equation method and the complete discriminant system for polynomial method, which are richer than the solutions acquired in existing papers. We draw the two-dimensional graphs of amplitudes and corresponding chirps in order to verify the existence of the solutions and discuss the dynamical properties of the solutions. To our knowledge, this is the first time that comprehensive set of exact chirped solutions of the governing equation in the paper are obtained. The model and the results obtained in this paper may help explain some nonlinear problems.

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

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.

Revisitation of “implicit quiescent optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion": generalized temporal evolution

AbstractThis paper is a revisitation to the study on the retrieval of quiescent optical solitons to the complex Ginzburg–Landau equation that is considered with generalized temporal evolution and nonlinear chromatic dispersion. The results are recovered with the application of Lie symmetry. Apart from a couple of self-phase modulation structures where the results are in quadratures, the integrals are with a range of special functions. Two new forms of self-phase modulation structures are addressed in the paper.

Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg–Landau equation

This research focuses on investigating an extended version of the Ginzburg–Landau (GL) equation that describes the motion of particles in a plasma. The other applications of this model can be found in optics, and other related fields. As part of our approach, we seek to discover some new wave solutions to the model, and then use the Galilean transformation in order to determine the model’s dynamical properties. After that, we use planar dynamical system theory to perform an in-depth bifurcation analysis, which gives us valuable insight into the system. By conducting thorough mathematical and bifurcation analyses, we identify the fundamental dynamics and critical points that regulate the system’s behavior. Moreover we explore the chaotic nature of the system, uncovering potential chaotic tendencies and their consequences. Further, we obtain several categories of optical solutions of the complex GL equation by utilizing new logarithmic transformations, offering valuable insight into its behavior and potential applications in optics. Our analytical technique yields closed-form solutions expressing elementary functions. In order to ensure the reliability of our findings, we rigorously validate the obtained solutions by substituting them back into the original model. Our research helps us better understand this equation’s characteristics and its relevance across a wide range of disciplines.

Efficient techniques for nonlinear dynamics: a study of fractional generalized quintic Ginzburg-Landau equation

In this research, we explore the solution of the fractional Generalized Quintic Complex Ginzburg-Landau equation (GQCGLE) using the Controlled Picard Transform Method. We solve the equation by incorporating the Laplace transform (LT) and the Caputo-Fabrizio fractional derivative (CF). The fractional GQCGLE is a complex-valued equation that exhibits intricate dynamics and plays a significant role in various physical phenomena, particularly in the realm of nonlinear optics. The suggested method can efficiently address a wide variety of differential equations, encompassing both integer and fractional-order equations. This is achieved by integrating an extra parameter that enhances convergence, making it especially suitable for non-linear differential equations. To confirm the precision and convergence of our approach, we validate it by comparing it with other methods, presenting graphical representations to clarify the effects of different parameters on the solution behaviour, and confirming its precision and convergence. The existence and uniqueness of the solutions are also examined.

Irreducibility of Stochastic Complex Ginzburg-Landau Equations Driven by Pure Jump Noise and Its Applications

Irreducibility of Stochastic Complex Ginzburg-Landau Equations Driven by Pure Jump Noise and Its Applications

Impact of the Higher-Order Reactive Nonlinearity on High-Amplitude Dissipative Solitons

In this work, the impact of the higher-order reactive nonlinearity on very high-amplitude solitons of the cubic–quintic complex Ginzburg–Landau equation is investigated. These high amplitude pulses were found in a previous work in the normal and anomalous dispersion regimes, starting from a singularity found by Akhmediev et al. We focus mainly in the normal dispersion regime, where the energy of such pulses is particularly high. In the presence of the higher-order reactive nonlinearity effect, pulse formation are observed for much higher absolute values of dispersion. Under such effect, the amplitude and the energy of the VHA pulses decrease, while their spectral range shrinks. Numerical computations are in good agreement with the predictions based on the method of moments, in the absence of the higher-order reactive nonlinearity effect. However, in the presence of this effect such agreement becomes mainly qualitative. A region of existence of the very high-amplitude pulses was found in the semi-plane defined by the normal dispersion and nonlinear gain.

Universality of oscillatory instabilities in fluid mechanical systems

Oscillatory instability emerges amidst turbulent states in experiments in various turbulent fluid and thermo-fluid systems such as aero-acoustic, thermoacoustic and aeroelastic systems. For the time series of the relevant dynamic variable at the onset of the oscillatory instability, universal scaling behaviors have been discovered in experiments via the Hurst exponent and certain spectral measures. By means of a center manifold reduction, the spatiotemporal dynamics of these real systems can be mapped to a complex Ginzburg–Landau equation with a linear global coupling. In this work, we show that this model is able to capture the universal behaviors of the route to oscillatory instability, elucidating it as a transition from defect to phase turbulence mediated by the global coupling.

High dispersion and cubic–quintic–septic–nonic nonlinearity effects on optical solitons in the complex Ginzburg–Landau equation of eighth-order with multiplicative white noise in the Itô sense

In this article, we present the complex Ginzburg–Landau equation involving eighth-order dispersion and multiplicative white noise effect in the context of Itô sense for the first time. In order to investigate and analyze the proposed model, two efficient and convenient techniques have been employed in this study. The results revealed the presence of various solitons, including bright, dark, and singular solitons. Additionally, the existence of straddled solitons was reported as well. These findings contribute to the current understanding of soliton dynamics and provide valuable insights for further research in this field. To better understand our findings, we have included several graphs that effectively illustrate the effect of the white noise. The white noise effect refers to a specific pattern of random fluctuations that can be observed in various systems and phenomena. It is characterized by a uniform distribution of frequencies across the entire spectrum, resulting in a constant and uncorrelated signal.

Emergence of spiral and antispiral patterns and its CGLE analysis in leech-heart interneuron model with electromagnetic induction

Neurons can exhibit various rhythmic activities such as bursting, spiking, and quiescent states when exposed to external input current stimulus. In this paper, a model of medicinal leech's heart (LH) interneuron is considered to describe the dynamics of neurons with a varied range of electrical activities. The crucial insights into the model's dynamics are explored in three different parameter regimes: phasic spiking, regular spiking, and bursting, based upon the codimension-one bifurcation of the model by considering VK2shift as a bifurcation parameter. The spatiotemporal dynamics of the model are explored by allowing 1D and 2D diffusion in the membrane voltage. The 1D diffusive system produces irregular bursting dynamics for the intermediate value of diffusion coefficients, whereas, at higher values, it shows synchronized oscillations. In the presence of 2D diffusion, the emergence of different types of spiral patterns is observed in the system. Furthermore, the system is extended by incorporating electromagnetic induction in the membrane voltage to explore the effect of induction on the various dynamics of neural model. By varying its intensities, the membrane voltage in the extended model produces a variety of discharge modes, such as periodic spiking, fast-spiking, resting, and spike-adding phenomena. In addition, the emergence of anti-spiral patterns in the extended model near subcritical Hopf bifurcation is analytically verified using the complex Ginzburg-Landau equation (CGLE). These findings demonstrate that the firing patterns vary based on the control parameters, and these variations contribute to our understanding of how the brain system transmits and processes the signals.

Oscillatory nonlinear thermal instability in nanoliquid under gravity modulation within Hele-Shaw cell

Abstract The effect of gravity-field modulation is investigated in a nano liquid-confined Hele-Shaw cell. This study aims to finish the work described in (S. N. Rai, B. S. Bhadauria, K. Anish, and B. K. Singh, “Thermal instability in nanoliquid under four types of magnetic-field modulation within Hele-Shaw cell,” Int. J. Heat Mass Transfer, vol. 145, no. 7, p. 072501, 2023) for oscillatory convection. The existence of the complex Ginzburg-Landau equation (CGLE) model is constrained by the requirement ω 2 > 0. The magnetic fluxes in the Hele-shaw cell are governed by CGLE with g-jitter. The quantity of heat-mass transfer is examined in the presence of a g-jitter. In addition, the findings of our research on transport analysis indicate that oscillatory mode is preferable to stationary mode. It is also found that the gravity-driven Hele-Shaw layer has lower transport properties. Further, the transport analysis is compared to previous research and shown to have improved results.

Highly dispersive optical soliton perturbation with Kerr law for complex Ginzburg–Landau equation

Highly dispersive optical soliton perturbation with Kerr law for complex Ginzburg–Landau equation

Insights into oscillator network dynamics using a phase-isostable framework.

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bifurcations and dynamics of the network, which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators. In this work, we consider the phase-isostable network equations for an arbitrary but finite number of identical coupled oscillators, obtaining conditions required for the stability of phase-locked states including synchrony. For the mean-field complex Ginzburg-Landau equation where the solutions of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and, therefore, employ this to investigate the dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes). We observe qualitative correspondence between results from numerical simulations of the full system and the phase-isostable description demonstrating that in both small and large networks, the phase-isostable framework is able to capture dynamics that the first-order phase description cannot.