Well-posedness and dynamics for the fractional Ginzburg-Landau equation

Pu, Xueke; Guo, Boling

doi:10.1080/00036811.2011.614601

Cited by 70 publications

(26 citation statements)

References 14 publications

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“…The strange attractors in several fractional chaotic systems have been observed by numerical simulation, such as the fractional-order Chua-type oscillators, the fractional Lorenz's system and the fractional financial system [16][17][18][19][20][21]. The existence of attractors of space fractional Ginzburg-Landau equations and the Heisenberg equations was proved in [22]. It is natural to study the dissipativity of fractional-order systems and the corresponding attractor.…”

Section: Introductionmentioning

confidence: 97%

Dissipativity and contractivity for fractional-order systems

Wang

Xiao

2014

Nonlinear Dyn

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This paper concerns the dissipativity and contractivity of the Caputo fractional initial value problems. We prove that the systems have an absorbing set under the same assumptions as the classic integerorder systems. This directly extends the dissipativity from integer-order systems to the Caputo fractionalorder ones. The fractional dissipativity conditions can be satisfied by many fractional chaotic systems and the systems from the spatial discretization of some timefractional partial differential equations. The fractionalorder systems satisfying the so-called one-sided Lipschitz condition are also considered in a similar way, and the contractivity property of their solutions is proved. Two numerical examples are provided to illustrate the theoretical results.

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Section: Introductionmentioning

confidence: 97%

Dissipativity and contractivity for fractional-order systems

Wang

Xiao

2014

Nonlinear Dyn

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“…There are many papers devoted to investigating the theoretical properties of FGLEs. For instance: Tarasov derived and analyzed the psi‐series solution; Pu and Guo investigated the global well‐posedness, long‐time dynamics and global attractors for the nonlinear FGLE; Li and Xia also considered the well‐posedness of real FGLE in Sobolev spaces; see for more references.…”

Section: Introductionmentioning

confidence: 99%

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

Hao

Sun

2016

Numerical Methods Partial

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The numerical solution for the one‐dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high‐order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap‐frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017

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“…Recently, fractional partial differential equations arise in a wide range of fields within in physics, biology, chemistry, etc., and some classical equations of mathematical physics have been postulated with fractional derivative to better describe complex phenomena, including the fractional Schrödinger equation [10,15,16], fractional Landau-Lifshitz equation [19], fractional Landau-Lifshitz-Maxwell equation [32] and fractional Ginzburg-Landau equation [17,33,42].…”

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confidence: 99%

“…The global existence and long behavior of Ginzburg-Landau equation were studied in [9,11,18,22,23]. In fractional case, the well-posedness and dynamical behavior were proved in [26,33]. For stochastic fractional Ginzburg-Landau equation with α ∈ ( 1 2 , 1), the existence of the random attractor in L 2 and H 1 were respectively discussed in [28,38,41].…”

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confidence: 99%

Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise

Lan¹,

Shu²

2019

Communications on Pure &Amp; Applied Analysis

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This paper is concerned with the asymptotic behavior of solutions for non-autonomous stochastic fractional complex Ginzburg-Landau equations driven by multiplicative noise with α ∈ (0, 1). We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions, which is slightly different from the deterministic fractional case with α ∈ (1 2 , 1) and the real fractional case with α ∈ (0, 1). Consequently, we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain. At last, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero.

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Well-posedness and dynamics for the fractional Ginzburg-Landau equation

Cited by 70 publications

References 14 publications

Dissipativity and contractivity for fractional-order systems

Dissipativity and contractivity for fractional-order systems

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise

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