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In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ -mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies. • Lawson and splitting methods of order 4 for Complex Ginzburg–Landau equations. • Efficient implementation by tensor-matrix products or FFT techniques. • Physical 2D and 3D numerical examples (cubic, cubic-quintic, and coupled CGL). [ABSTRACT FROM AUTHOR]

We present weighted estimates and higher order asymptotic expansions of global solutions to the complex Ginzburg–Landau (CGL) type equation in the super Fujita-critical case. Our approach is based on commutation relations between the CGL semigroup and monomial weights in $ \mathbb{R}^{n} $ for the weighted estimates and on the Taylor expansions with respect to the both space and time variables for the asymptotic expansions. We also characterize the optimal decay rate in time of the remainder for the asymptotic expansion from the viewpoint of the moments of the initial data in space and those of the nonlinear term in spacetime. [ABSTRACT FROM AUTHOR]

When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media. [ABSTRACT FROM AUTHOR]

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the complex Ginzburg-Landau equation on the torus when the core size of vortex ε → 0. The reduced dynamical law of the complex Ginzburg-Landau equation is governed by a mixed flow of gradient flow and Hamiltonian flow which are both driven by a renormalized energy on the torus. Finally, some first integrals and analytic solutions of the reduced dynamical law are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This study examines the analytic wave solutions of a highly dispersive perturbed complex Ginzburg–Landau equation (CGLE) with conformable fractional derivative and polynomial law of nonlinearity using the improved modified extended tanh-function method. The results show a wide range of solutions including (bright, dark, singular) solitons, Jacobi elliptic solutions, exponential solutions, and Weierstrass elliptic solutions. The obtained soliton solutions showcase diverse dynamics, encompassing different solitary waves and localized structures. The polynomial nonlinearity adds complexity to the dynamics, resulting in the emergence of new solitons with distinct characteristics. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on the complex Ginzburg–Landau equation, propagation dynamics of multipole solitons generated in the dissipative system are numerically investigated by the split-step Fourier method. The effect of the value of the different Lévy indexes on stability regions of the soliton has been explored. In addition, we observe domains of different outcomes of the evolution of the input beam in the parameter plane of linear loss coefficient or diffraction gain coefficient and cubic gain coefficient. The results show that the evolution can lead to three different outcomes: decay, development into stable single soliton, expansion into the spreading pattern. We also study the evolution of multipole solitons generated with larger quintic loss coefficients and find that the input splits into the symmetrical fragments in the initial propagation. It is also demonstrated that two solitons or three solitons merge into the single soliton. Meanwhile, the relationship of merging distance with Lévy index and initial amplitude is also given. [ABSTRACT FROM AUTHOR]

In this paper, we go deeply into the complex Ginzburg–Landau equation with highly dispersive perturbed birefringent fibers having a polynomial law of nonlinearity and acquire three modes of solutions, including solitary wave modes, singular modes, and elliptic function double periodic modes, by using the trial equation method and the complete discrimination system for polynomials. In order to digest the dynamic properties of the model better, we study accurate two-dimensional and three-dimensional images of solutions at specific values. The study of this equation is of great significance for the research and application of superconductors. [ABSTRACT FROM AUTHOR]

We present an explicit derivation of a (3+1)-dimensional [(3+1)D] cubic- quintic-septic complex Ginzburg–Landau (CQS-CGL) equation, including diffraction, linear dispersions up to the seventh order, loss, gain, cubic-quintic-septic nonlinearities, as well as cubic-quintic-septic first-order self-steepening effects. The new model equation, derived from Maxwell equations beyond the slowly varying envelope approximation, describes the dynamics of dissipative light bullets in nonlinear metamaterials (MMs). Using direct numerical simulations of the whole (3+1)D CQS-CGL equation, we present the evolution of various dissipative optical bullets in MMs characterized by different topological charges, namely, the fundamental vortex, necklace, and azimuthons. The bullet amplitudes and phase distributions support the emergence of new propagating modes under parameter values that promote their instability. However, with the right choice of higher-order parameters, especially the cubic, quintic and septic self-steepening coefficients, the numerical simulations are capable of achieving the stability of the studied. Under unstable conditions, even multipole vortices are found to converge in the rotating frame, the fundamental spherical light bullet, while their amplitude drops drastically. The results suggest that the presence of higher-order nonlinear effects, balanced by the higher-order dispersive terms, prevent the light bullets, with different topological charges, from collapsing, with rotation direction specific to negative-index MMs. [ABSTRACT FROM AUTHOR]

The generalized complex Ginzburg–Landau equation is considered. An analytical condition for the existence of horseshoe chaos is obtained for the traveling wave reduction of the investigated equation by using the Melnikov method. A way to control chaos in the dynamical system is proposed. An analytical prediction is tested numerically by plotting attraction basins of attraction of the Poincare section of the studied system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional complex Ginzburg–Landau equation with Kerr law and power law nonlinearity. Using the conformable derivative approach and the bifurcation method, we effectively derived new explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and antikink wave solution, compacton) under different parameter conditions. The quasiperiodic, chaotic behavior and sensitivity analysis of the model is studied for different values of parameters after deploying an external periodic force. Finally, various 2D and 3D simulation figures are plotted to show the physical significance of these exact solutions. [ABSTRACT FROM AUTHOR]

A cubic–quintic complex Ginzburg–Landau equation that describes the dynamics of the field in passively mode-locked lasers is considered. For passively mode-locked lasers with spectral filtering, we show analytically that the competing cubic–quintic nonlinearity induces propagating dissipative chirp-free solitonlike dark/bright solitons and double-kink solitons in our physical model. In the case of model without spectral filtering, the phase-engineering technique is employed to produce approximate bright and dark soliton solutions for investigating the dynamics of chirped femtosecond dissipative solitons in passively mode-locked lasers under consideration. Parameter domains are delineated in which the chirp-free and chirped pulses exist. We show that the nonlinear chirp associated with each pulse in the case of absence of spectral filtering is directly proportional to the intensity of the wave. Also, we show that the amplitude and the width of pulses and those of the corresponding chirping can be controlled by varying various parameters such as, for example, the linear parabolic gain dispersion, the nonlinear gain, the linear gain/loss, the nonlinear chirping parameter. Our theoretical results are confirmed by direct numerical simulations on the model equations. [ABSTRACT FROM AUTHOR]

This paper obtains the stationary optical solitons and new exact solutions for complex Ginzburg–Landau equation (CGLE) with nonlinear chromatic dispersion and Kudryashov's reflective index structure in a non-Kerr law media. The research work is carried out according to the following route: first, the CGLE is transformed into a second order nonlinear ordinary differential equation by an appropriate substitution. Then, the bifurcation, stationary soliton solution and exact solution of CGLE are obtained by using dynamic system theory and polynomial complete discriminant system method. Abundant solutions are obtained, including periodic solutions, doubly-periodic solutions, hyperbolic function solutions, rational function solutions and exponential function solutions. Finally, the 3D and 2D graphics for the solutions are drawn. Since the appearance of stationary soliton means the stop of signal transmission, the research results provide a way to avoid the disaster of superconducting propagation in nonlinear media. [ABSTRACT FROM AUTHOR]

The fractional complex Ginzburg–Landau model is widely used in the description of wave propagation through optical transmission lines. It has many useful applications in the fields of telecommunications and nonlinear optics. This paper investigates the fractional effects of the complex Ginzburg–Landau model with quadratic–cubic, anti–cubic and generalized anti–cubic laws of nonlinearity by using generalized projective Riccati equation method. The variation in the traveling wave behavior of the governing model is examined for beta, conformable and M-truncated derivatives. Some constraint conditions are carried out during mathematical analysis, which are further used for evaluating the traveling wave solutions. The analytic solutions of the considered model are determined in terms of hyperbolic and trigonometric function solutions. Consequently, dark, bright, kink, bell-shaped and singular solitons are extracted. The reported solutions are presented using 2D and 3D graphs. These graphs are showing the fractional effects for different values of fractional parameter. The evolution of the wave profiles shows that the retrieved solitons become similar for all three definitions of fractional derivatives as the fractional parameter approaches unity.

In this paper, the cubic–quartic complex Ginzburg–Landau (CGL) equation is investigated by using the trial function method. The traveling wave hypothesis is applied to convert the CGL equation to an ordinary differential equation (ODE), which is equivalent to a dynamic system. The qualitative behavior, bifurcation of the phase portraits for this system is studied. Furthermore, the chaotic motions in system involving external periodic perturbation are considered. Some new optical solitons, such as Jacobian elliptic function solutions, for CGL equation with quadratic–cubic law nonlinearity are constructed using the complete discrimination system for polynomial method. Moreover, two-dimensional graphs, three-dimensional, corresponding contour and density plots for some acquired solutions are depicted to carry out the physical properties of optical pulse propagation.

Ginzburg-Landau equation has two types of behavior: one is spatio-temporal chaos lying inside the limit cycle on the two dimensional space, the other is a spatially homogeneous periodic solution on the limit cycle. If we perturb the solution behaving spatio-temporal chaos to the outside of a limit cycle, it is numerically observed that the perturbed solution converges to a spatially homogeneous periodic oscillation. This is the transition from chaos to regular motions based on a spatial homogenization by the perturbation. By constructing the invariant sets and using the asymptotic stability of the limit cycle, we prove analytically that the solution starting from an initial condition far from the limit cycle converges to the limit cycle oscillation. [ABSTRACT FROM AUTHOR]

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.

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well. [ABSTRACT FROM AUTHOR]

This article considers the complex Ginzburg–Landau equation which arises in many fields of science and engineering, specifically in nonlinear fiber optics. Three different forms of nonlinearity, namely, the Kerr law, quadratic-cubic law and parabolic law are discussed for the proposed model. The governing equation is explored using the generalized projective Riccati equations technique. A variety of soliton solutions are extracted that describe the dynamic structure of the model. The results are illustrated graphically using 3D graphs and 2D contour plots that assist in examining the solitonic behavior of the solutions for suitable choices of arbitrary parameters. [ABSTRACT FROM AUTHOR]

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 ϕ6-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. [ABSTRACT FROM AUTHOR]

A new data-driven reduced order model is developed to efficiently simulate transient dynamics, with the aim at computing the final attractor. The method combines a standard numerical solver and time extrapolation based on a recent data processing tool, called higher order dynamic mode decomposition. Such combination is made using interspersed time intervals, called snapshots computation (obtained from the numerical solver) and extrapolation intervals (computed by the model). The process continues, alternating snapshots computation and extrapolation intervals, until that moment at which the final attractor is reached. Thus, the method adapts the extrapolated approximations to the varying dynamics along the transient simulation. The performance of the new method is tested considering representative transient solutions of the one-dimensional complex Ginzburg–Landau equation. In this application, the speed-up factor of the reduced order model, comparing its computational cost with that of the standard numerical solver, is always much larger than one in the computation of representative periodic and quasi-periodic attractors. [ABSTRACT FROM AUTHOR]

In this article, the recently discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction–diffusion partial differential equations (PDEs) is analysed in the cubic Complex Ginzburg–Landau equation, as an equation in its own right, with a slowly varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents on the linearized PDE to show that solutions, which have approached the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linearized PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The non-linear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS in the full, non-linear PDE. Next, we show that beyond a key Stokes line through the saddle there is a curve in the space-time plane along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the non-linear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. This curve is called the space–time buffer curve. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial data and time. Hence, a competition arises between these two solutions, as to which one ceases to be exponentially small first, and this competition governs spatial dependence of the DHB. We find four different cases of DHB, depending on the outcomes of the competition, and we quantify to leading order how these depend on the main system parameters, including the Hopf frequency, initial time, initial data, source terms, and diffusivity. Examples are presented for each case, with source terms that are a uni-modal function, a smooth step function, a spatially periodic function and an algebraically growing function. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence, region-specific control over the delayed onset of oscillations can be achieved. [ABSTRACT FROM AUTHOR]

The current work is on the retrieval of stationary soliton solutions to the complex Ginzburg–Landau equation that is studied with nonlinear chromatic dispersion having a plethora of nonlinear refractive index structures. The Jacobi's elliptic function approach is employed to recover doubly periodic waves which leads to soliton solutions when the limiting value of the modulus of ellipticity is reached. [ABSTRACT FROM AUTHOR]

In recent years, nonlinear concepts have attracted a lot of attention due to the deep mathematics and physics they contain. In explaining these concepts, nonlinear differential equations appear as an inevitable tool. In the past century, considerable efforts have been made and will continue to be made to solve many nonlinear differential equations. This study is also a step towards analytical solution of the complex Ginzburg-Landau equation (CGLE) used to describe many phenomena on a wide scale. In this study, the CGLE was solved analytically by $(1/G')$-expansion method.

Abstract In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency.

We consider an initial boundary value problem of the complex Ginzburg-Landau equation with some delayed feedback terms proposed for the control of chemical turbulence in reaction diffusion systems. We consider the equation in a bounded domain $\Omega\subset\mathbb{R}^{N}$ ($N\leq3$), $$ \frac{\partial u}{\partial t}-(1+i\epsilon)\Delta u +(1+i\beta) | u| ^2u-(1-i\omega) u=F(u(x,t-\tau)) $$ for t>0, with $$ F(u(x,t-\tau)) =e^{i\chi_0}\big\{ \frac{\mu}{| \Omega| }\int_{\Omega}u(x,t-\tau) dx+\nu u(x,t-\tau) \big\} , $$ where $\mu$, $\nu\geq0$, $\tau>0$ but the rest of real parameters $\epsilon$, $\beta$, $\omega$ and $\chi_0$ do not have a prescribed sign. We prove the existence and uniqueness of weak solutions of problem for a range of initial data and parameters. When $\nu=0$ and $\mu>0$ we prove that only the initial history of the integral on $\Omega$ of the unknown on $(-\tau,0)$ and a standard initial condition at t=0 are required to determine univocally the existence of a solution. We prove several qualitative properties of solutions, such as the finite extinction time (or the zero exact controllability) and the finite speed of propagation, when the term $|u| ^2u$ is replaced by $|u| ^{m-1}u$, for some $m\in(0,1)$. We extend to the delayed case some previous results in the literature of complex equations without any delay.

This paper secures exact solutions from perturbed complex Ginzburg–Landau equation that is taken into account with Kerr law and cubic–quintic–septic nonlinearity. Two approaches are used, namely the trial equation method and complete discriminant system for polynomial method. The abundant exact solutions obtained can better analyze the complex optical phenomena and further demonstrate their essence.

This article examines the complex Ginzburg–Landau equation with the beta time derivative and analyze its optical solitons and other solutions in the appearance of a detuning factor in non-linear optics. The kink, bright, W-shaped bright, and dark solitons solution of this model are acquired using the modified Exp-function and Kudryshov methods. The model is examined with quadratic-cubic law, Kerr law, and parabolic laws non-linear fibers. These solitons emerge with restrictive conditions that ensure their existence are also presented. Furthermore, the obtained and precise solutions are graphically displayed, illustrating the impact of non-linearity. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. [ABSTRACT FROM AUTHOR]

Femtosecond lasers interacting with Kerr nonlinear optical materials, propagate in form of filaments due to the balance of beam diffraction by self-focusing induced by the Kerr nonlinearity. Femtosecond laser filamentation is a universal phenomenon that belongs to a general class of processes proper to ultrashort lasers processing systems, associated with the competition between nonlinearity and dispersion also known to promote optical solitons. The present work considers a model describing femtosecond laser inscriptions in a transparent medium with Kerr nonlinearity. Upon inscription, the laser stores energy in the optical material which induces an electron plasma. The model consists of a cubic complex Ginzburg-Landau equation, in which an additional K-order nonlinear term takes into account K-photon absorption processes. The complex Ginzburg-Landau equation is coupled to a time first-order nonlinear ordinary differential equation, accounting for time evolution of the plasma density. The main objective of the study is to examine effects of the competition between multi-photon absorptions, radiative recombination and electron diffusion processes, on temporal profiles of the laser amplitude as well as of the plasma density. From numerical simulations, it is found that when the photon number (i.e. K) contributing to multiphoton ionization is large enough, taking the electron diffusion processes into account favours periodic structures in temporal profiles both of the laser and the plasma density. The pulse repetition rate in the optical soliton train is increased with increase of the electron diffusion coefficient, while the plasma density is a train of multi-periodic anharmonic wave patterns. [ABSTRACT FROM AUTHOR]

This article obtains the optical solitons of the complex fractional Ginzburg–Landau equation by the hypothesis of traveling wave and generalized projective Riccati equation scheme. There are four conditions, Kerr law, parabolic law, power law and dual power law of nonlinearity associated with the model. The constraint conditions for the existence of these solutions have also been discussed. Moreover, the physical significance of the constructed solutions has been provided using graphical representation. A comparative study is made by using two distinct definitions of fractional derivatives namely as Beta and M-truncated. Furthermore, a quantitative overview is also included, which involves solutions to the model under discussion. The complex Ginzburg–Landau equation is subjected to a comprehensive sensitivity analysis. Finally, the modulation instability (MI) analysis of proposed model is also carried out on the basis of linear stability analysis. A dispersion relation is obtained between the wave number and frequency.

A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which are selected via Gauss elimination with double pivoting on the original snapshot matrix containing the given data. In the present work, the sampling method is adapted and combined with low-dimensional modeling. This combination yields a novel adaptive algorithm for the simulation of time-dependent non-linear dynamics in pattern-forming systems. Convenient snapshot sets, computed on demand over the evolution, are stored to record local temporal events whose underlying mechanisms are essential for the approximations. Also, a collection of sparse grid points, which are used to construct the mode basis and the reduced system of equations, is adaptively sampled according to unlinked spatial structures. The outcome is a reduced order model of the problem that (i) yields reliable approximations of the dynamical transitions, (ii) is well-suited to describe localized spatio-temporal complexity, and (iii) provides fast computations. Robustness, accuracy, and computational efficiency of the proposed algorithm are illustrated for some relevant pattern-forming systems, in both one and two spatial dimensions, exhibiting solutions with a rich spatio-temporal structure. [ABSTRACT FROM AUTHOR]

In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg–Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed to establish linear algebraic systems. In the method, the penalty technique is used to facilitate the satisfaction of boundary conditions, and the stabilized moving least squares approximation is used to enhance the stability and performance. The linearized EFG method is a meshless method and possesses high precision and convergence rate in both space and time. Theoretical error and convergence of the linearized EFG method are analyzed. Finally, some numerical results are provided to demonstrate the efficiency of the method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The complex Ginzburg–Landau (CGL) equation which describes the soliton propagation in the presence of the detuning factor is firstly considered; then its solitons as well as Jacobi elliptic function solutions are obtained systematically using a modified Jacobi elliptic expansion method. In special cases, several dark and bright soliton solutions to the CGL equation are retrieved when the modulus of ellipticity approaches unity. The results presented in the current work can help to complete previous studies on the complex Ginzburg–Landau equation.

We prove the existence of m-armed spiral wave solutions for the complex Ginzburg– Landau equation in the circular and spherical geometries. We establish a new global bifurcation approach and generalize the results of existence for rigidly rotating spiral waves. Moreover, we prove the existence of two new patterns: frozen spirals in the circular and spherical geometries, and 2-tip spirals in the spherical geometry. [ABSTRACT FROM AUTHOR]

In this research article, our aim is to construct new optical soliton solutions for nonlinear complex Ginzburg–Landau equation with the help of modified mathematical technique. In this work, we studied both laws of nonlinearity (Kerr and power laws). The obtained solutions represent dark and bright solitons, singular and combined bright-dark solitons, traveling wave, and periodic solitary wave. The determined solutions provide help in the development of optical fibers, soliton dynamics, and nonlinear optics. The constructed solitonic solutions prove that the applicable technique is more reliable, efficient, fruitful and powerful to investigate higher order complex nonlinear partial differential equations (PDEs) involved in mathematical physics, quantum plasma, geophysics, mechanics, fiber optics, field of engineering, and many other kinds of applied sciences. [ABSTRACT FROM AUTHOR]

We prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations (these equations may be ill-posed): a single specific solution, an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions. In the later cases, which include several orbits, we also establish the invariant manifolds of an orbit depend smoothly on the orbit (analytically in the case of quasi-periodic orbits and finitely differentiably in the case of more general families). We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, analyticity) assumptions on the linearized equation, establishes the existence of the desired manifold. Related results appear in the literature, but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds. This makes the results in this paper applicable to some several models of current interest that could not be treated otherwise. We discuss in detail the Boussinesq equation of water waves (similar phenomena happen in other long wave approximations) and complex Ginzburg-Landau equation. More recently, we observed [11] that our results also apply to Mean Field Games. Since the equations we consider may be ill-posed, part of the requirements for the stable manifold is that one can define the (forward) dynamics on them. Note also that the methods that are based in the existence of dynamics (such as graph transform) do not apply to ill-posed equation. We use the methods based on integral equations (Perron method) associated with the partial dynamics, but we need to take advantage of smoothing properties of the partial dynamics. Note that, even if the families of solutions we started with are finite dimensional, the stable manifolds may be infinite dimensional. [ABSTRACT FROM AUTHOR]

We consider an initial boundary value problem of the complex Ginzburg-Landau equation with some delayed feedback terms proposed for the control of chemical turbulence in reaction diffusion systems. We consider the equation in a bounded domain Ω ⊂ ℝN (N ≤ 3), ... where μ, ν ≥ 0, τ > 0 but the rest of real parameters ε, β, ω and χ0 do not have a prescribed sign. We prove the existence and uniqueness of weak solutions of problem for a range of initial data and parameters. When ν = 0 and μ > 0 we prove that only the initial history of the integral on Ω of the unknown on (-τ, 0) and a standard initial condition at t = 0 are required to determine univocally the existence of a solution. We prove several qualitative properties of solutions, such as the finite extinction time (or the zero exact controllability) and the finite speed of propagation, when the term |u|²u is replaced by |u|m-1u, for some m ∈ (0, 1). We extend to the delayed case some previous results in the literature of complex equations without any delay. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency. [ABSTRACT FROM AUTHOR]

Blow-up phenomena of weakly coupled systems of several evolution equations, especially complex Ginzburg-Landau equations is shown by a straightforward ODE approach, not by the so-called test-function method used in [38] which gives the natural blow-up rate. The difficulty of the proof is that, unlike the single case, terms which come from the Laplacian cannot be absorbed into the weakly coupled nonlinearities. A similar ODE approach is applied to heat systems by Mochizuki [32] to obtain the lower estimate of lifespan.