Stochastic Complex Ginzburg-Landau Equation Research Articles

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

Averaging principle for stochastic complex Ginzburg-Landau equations

Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale ε goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as ε goes to zero.

Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noise

This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landauequation driven by locally Lipschitz nonlinear diffusion terms defined on $ R^n $. We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation.

Asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains

This paper mainly investigates the asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains. We first prove the existence and uniqueness of random attractors for the considered equation and its limit equation. Due to the non-compactness of Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of such a stochastic equation is proved by the tail-estimate method. Then, we show the upper semi-continuity of random attractors when thin domains collapse onto the real space R.

Dark-singular combo optical solitons with fractional complex Ginzburg–Landau equation

This paper employs dextended Jacobi's elliptic function expansion method to retrieve doubly periodic function as solutions to the stochastic complex Ginzburg–Landau equation. In the limiting case, when the modulus of ellipticity approaches unity, these solutions approach optical solitons. This paper lists the dark-singular combo optical solitons.

Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains

<p style='text-indent:20px;'>In this paper, we investigate the asymptotic behavior for non-autonomous stochastic complex Ginzburg-Landau equations with multiplicative noise on thin domains. For this aim, we first show that the existence and uniqueness of random attractors for the considered equations and the limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse onto an interval.

Stochastic complex Ginzburg-Landau equation with space-time white noise

We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

ABSTRACTThe existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.

A weighted identity for stochastic partial differential operators and its applications

In this paper, a weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in establishing Carleman-type estimates for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce some known global Carleman estimates for stochastic parabolic equations, stochastic Schrödinger equations, stochastic transport equations and their deterministic counterparts. Meanwhile, as its applications, we derive two different Carleman estimates for linear forward stochastic complex Ginzburg–Landau equations. They can be used to study the controllability/observability and inverse problems for some stochastic complex Ginzburg–Landau equations, respectively.

Scaling properties of one-dimensional driven-dissipative condensates

We numerically investigate the scaling properties of a one-dimensional driven-dissipative condensate described by a stochastic complex Ginzburg-Landau equation (SCGLE). We directly extract the static and dynamical scaling exponents from the dynamics of the condensate's phase field, and find that both coincide with the ones of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation. We furthermore calculate the spatial and the temporal two-point correlation functions of the condensate field itself. The decay of the temporal two-point correlator assumes a stretched-exponential form, providing further quantitative evidence for an effective KPZ description. Moreover, we confirm the observability of this non-equilibrium scaling for typical current experimental setups with exciton-polariton systems, if cavities with a reduced $Q$ factor are used.

Poster #M231 TWO NOVEL KV3 ION CHANNEL MODULATORS ALLEVIATE COGNITIVE DYSFUNCTION AND SOCIAL BEHAVIOUR DEFICITS OF RELEVANCE TO SCHIZOPHRENIA IN AN ANIMAL MODEL

This paper employs dextended Jacobi's elliptic function expansion method to retrieve doubly periodic function as solutions to the stochastic complex Ginzburg–Landau equation. In the limiting case, when the modulus of ellipticity approaches unity, these solutions approach optical solitons. This paper lists the dark-singular combo optical solitons.

Ergodicity for the stochastic Complex Ginzburg–Landau equations

We study a stochastic Complex Ginzburg–Landau (CGL) equation driven by a smooth noise in space and we establish exponential convergence of the Markov transition semi-group toward a unique invariant probability measure. Since Doob Theorem does not seem not to be useful in our situation, a coupling method is used. In order to make this method easier to understand, we first focus on two simple examples which contain most of the arguments and the essential difficulties.

Global Solution for a Stochastic Ginzburg-Landau Equation with Multiplicative Noise

This article is about stochastic complex Ginzburg-Landau equations with a multiplicative white noise. Existence of global solutions are proved in the same conditions as in the deterministic case.

Temporal localization of limit cycles in a noise-driven chemical oscillator

We study the influence of multiplicative random external perturbation on the Selkov model exhibiting limit cycles. The dynamics are described by stochastic differential equations. The numerically computed ensemble averages of the concentrations exhibit temporal localization of the limit cycles for small amplitudes of the noise. The localization time is smaller when the amplitude of the noise is higher. To understand this behavior, a stochastic complex Ginzburg–Landau equation is derived using normal form transformations on the stochastic evolution equations. In the case of weak noise, the equations for the radial and the phase components of the amplitude separate and can be solved exactly. The solution is used to compute the damping factors, which were found to be Gaussian. We also show that asymptotically, the radial part fluctuates near the average value and the phase is found to be a Gaussian random variable with time-dependent mean and variance and may be responsible for the temporal localization of the limit cycle.

Stability and noise in Taylor-Couette flow with through-flow

We present results of experimental and numerical work on a Taylor-Couette system with imposed axial flow. When the base flow is convectively unstable, macroscopic patterns of travelling Taylor vortices are observed downstream of the inlet. Numerical integration of the stochastic complex Ginzburg-Landau equation indicates that these patterns arise from the spatial amplification of microscopic noise. The noise leads to an irregular phase in the vortex pattern. In contrast, the phase is very regular when the base flow is absolutely unstable.