Generalized Ginzburg-Landau Equation Research Articles

A Large Deviation Principle for the Stochastic Generalized Ginzburg-Landau Equation Driven by Jump Noise

In this paper, we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise. The main difficulties come from the highly non-linear coefficient and the jump noise. Here, we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle, which was initially proposed by Matoussi, Sabbagh and Zhang (2021).

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Nonconservation of Topological Charge and Cusps in a One-Dimensional Laser Scheme.

An analysis was made of the field structures in a coaxial laser with fast saturable absorption. In such an efficiently one-dimensional scheme, the integer topological index (charge) naturally arises: the phase incursion of the field envelope when bypassing the aperture, divided by 2π. The topological charge is the winding number for the phase curve introduced for the electric field envelope with increase of the coordinate for fixed time. In the framework of the generalized Ginzburg-Landau equation, shown are finite numbers of plane-wave modes, stable with respect to weak perturbations, and solitonlike modes with inhomogeneous intensity distributions and different topological charge. It was found that the topological charge could change during transient to steady state. The events of these changes alternate with formation of cusps of the phase curve occurring when extrema of the field intensity and phase coincide.

Laser solitons: topological and quantum phenomena

A review of the properties of dissipative solitons with different dimensions and various topological characteristics in lasers and laser systems with saturable absorption is presented. Unlike conservative solitons, laser solitons are attractors, the increased stability of which is caused by the balance of energy inflow and outflow. The topology of laser solitons is due to their complex internal structure, which is determined by the field of radiation energy fluxes, and the energy characteristics are an important addition to the topological characteristics. The equation of their dynamics — the generalized Ginzburg–Landau equation — reflects the basic features of open nonlinear systems of various natures. The topological features of solitons expand the range of manifestations of their quantum fluctuations.

A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain {}{nonlinear} parabolic PDEs with boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction-diffusion PDE and a generalized Ginzburg-Landau equation with {}{mixed} boundary disturbances. Compared to some existing methods, the scheme proposed in this paper involves less intensive computations and can be applied to the ISS analysis for a {wide} class of nonlinear PDEs with boundary disturbances.

Numerical Study on Supercontinuum Generation in an Active Highly Nonlinear Photonic Crystal Fiber With Anomalous Dispersion

We numerically study supercontinuum (SC) generation (SCG) in a rare-earth-doped highly nonlinear photonic crystal fiber (HNL-PCF) with anomalous dispersion (AD) in the sub-picosecond pulse regime. We develop a semi-classical numerical model based on the generalized Ginzburg-Landau equation in order to take account of ultrafast interactions between gain ions and ultra-broadband SC radiation encompassing sub-100-femtosecond solitons. Based on the numerical model, we analyze SCG characteristics of an active HNL-PCF in comparison with a passive-type counterpart, unveiling novel optical gain effects in a highly nonlinear optical fiber with AD. We rigorously investigate gain-induced soliton dynamics, such as soliton-cascade-like behaviors, soliton-quasi-soliton collisions, and phase-matched dispersive wave generation, which eventually contributes to enhancement of energy scaling of SC radiation without incurring considerable degradation of its spectral flatness. We also verify that such superior performance characteristics of an active HNL-PCF make it suitable for the use as a boost amplifier for SC radiation. We think that the findings from this study will incite other subsequent studies on unveiling detailed nonlinear pulse dynamics in various gain-embedded nonlinear optical media.

Topological transformations of three-dimensional dissipative solitons in the framework of the generalized Ginzburg—Landau equation

We classify and analyze transformations of three-dimensional dissipative tangle-solitons, i.e., solitons with closed and nonclosed vortex lines, under smooth variations of the system parameters. An example of such a system is a laser medium with fast saturable absorption. We find several scenarios of both reversible and irreversible transformations and discuss the role of the dissipative property of the system (openness) in specific features of these phenomena.

Unconditional superconvergence analysis of a linearized Crank–Nicolson Galerkin FEM for generalized Ginzburg–Landau equation

In this paper, we study an effective numerical method for the generalized Ginzburg–Landau equation (GLE). Based on the linearized Crank–Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space–time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order O(h2) in the sense of L2-norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order O(τ2+h2) in the sense of H1-norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis.

A Stochastic Generalized Ginzburg–Landau Equation Driven by Jump Noise

This paper is concerned with the stochastic generalized Ginzburg-Landau equation driven by a multiplicative noise of jump type. By a prior estimate, weak convergence and monotonicity technique, we prove the existence and uniqueness of the solution of an initial-boundary value problem with homogeneous Dirichlet boundary condition. However, for the generalized Ginzburg-Landau equation, such a locally monotonic condition of the nonlinear term can not be satisfied in a straight way. For this, we utilize the characteristic structure of nonlinear term and refined analysis to overcome this gap.

Retraction Note to: Some blow-up results for a generalized Ginzburg-Landau equation

We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in φ defined on the whole space Rn, n = 1 or 2.

Inviscid limit behavior of the solution to the one-dimensional generalized Ginzburg–Landau equation

The Cauchy problem for the one-dimensional generalized Ginzburg–Landau (GGL) equation is considered and the inviscid limit behavior of its solution is proved. That is, the solution of Cauchy problem for the GGL equation converges to the solution of Cauchy problem for the derivative nonlinear Schrödinger equation in the natural space C([0,T];Hs) with s>12, for some T>0, if some coefficients tend to zero. Moreover, the convergence holds in C([0,T];H1) for any T>0 if initial data belong to H2. Furthermore, the convergence also holds in C([0,T];Hs) for any T>0 if initial data belong to Hs(s>47) with some coefficients conditions.

Families of solutions to the generalized Ginzburg-Landau equation and structural transitions between them

Solutions to the generalized Ginzburg-Landau equations for superconductors are obtained for a Ginzburg-Landau parameter κ close to unity. The families of solutions with arbitrary number n of flux quanta in a unit cell are analyzed. It is shown that under certain conditions, a cascade of phase transitions between different structures in a magnetic field appears near Tc. Algebraic equations are derived for determining the boundaries of coexistence of different phases on the {T, H0} plane.

Nanostructure Formation of the Solid Surface under the Action of Intense Laser Radiation

The paper describes the mechanism for formationof periodic nanostructures on the surface of solidsunder the action of intense laser radiation. Themodel based on a nonlinear relaxation oftemperature-induced stresses. The solid deformedby laser radiation is considered in a form of threesimultaneously coexisting phases. The system ofequations, which describes these phases, is reducedto the generalized Ginzburg–Landau equation thathas a periodic solution. For estimation thecalculation of the step of the periodic structurearising on the silicon surface under the action of alaser pulse is done.

Quasi-Periodic Solutions for the Generalized Ginzburg-Landau Equation with Derivatives in the Nonlinearity

In this paper, we discuss the existence of time quasi-periodic solutions for the generalized Ginzburg-Landau equation under periodic boundary conditions. By constructing a KAM theorem for dissipative systems with unbounded perturbations and multiple normal frequencies, we obtain a Cantorian branch of 2-dimensional invariant tori for the generalized Ginzburg-Landau equation.

Self-organization of dissipationless solitons in positive- and negative-refractive-index materials

General nonlinear and nonparaxial dissipative complex Helmholtz equations for magnetic and electric fields propagating in negative refractive index materials (NIMs) are derived ab initio from Maxwell equations. In order to describe nonconservative soliton dynamics in NIMs, such coupled equations are reduced into generalized Ginzburg-Landau equation. Cross-compensation between the excess of saturating nonlinearity, losses, and gain renders these self-organized solitons dissipationless and exceptionally robust. The presence of such solitons makes NIMs effectively dissipationless.

Generalized Hermite Spectral Method and its Applications to Problems in Unbounded Domains

In this paper, we develop a spectral method based on generalized Hermite functions with weight $\chi(x)\equiv1$. We also establish some basic results on generalized Hermite orthogonal approximations, which play an important role in spectral methods. As examples, the generalized Ginzburg-Landau equation in a population problem and an elliptic equation with a harmonic potential are considered. Related spectral schemes are proposed, and their convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.

Послекритические и докритические бифуркации бегущих волн модифицированного уравнения Гинзбурга–Ландау

Послекритические и докритические бифуркации бегущих волн модифицированного уравнения Гинзбурга–Ландау

Complexes of moving transversely one-dimensional laser solitons

Complexes of weakly coupled moving solitons are found by numerical solution of the generalized Ginzburg-Landau equation for the transversely one-dimensional scheme of a wide-aperture A-class laser with saturable absorption. As distinct from the case of complexes of motionless solitons, the phase difference between the neighboring moving solitons is close to 7π/2.

The attractor of the stochastic generalized Ginzburg-Landau equation

The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in H01.

A Dirichlet inhomogeneous boundary value problem for a generalized Ginzburg–Landau equation

The Ginzburg–Landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study a generalized complex Ginzburg–Landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives. We prove a global existence and uniqueness theorem.