Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

Guo, Boling; Huo, Zhaohui

doi:10.2478/s13540-013-0014-y

Cited by 69 publications

(33 citation statements)

References 11 publications

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“…When γ = 0 and α ∈ (0, 1), the problem (1.1) is a nonlocal model known as nonlinear fractional Schrödinger equation which has also attracted much attentions recently [9,10,11,12,19,20,21,22,23,24,15,16]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [29,30] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths.…”

Section: (T) = M (U(t)) :=mentioning

confidence: 99%

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Saanouni

2016

Journal of Mathematical Physics

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Abstract. Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated. Consider the initial value problem for an inhomogeneous nonlinear Schrödinger equation Contentswhich models various physical contexts in the description of nonlinear waves such as propagation of a laser beam and plasma waves. For example, when γ = 0, it arises in nonlinear optics, plasma physics and fluid mechanics [2,3]. When γ > 0, it can be thought of as Date: February 16, 2016. 1991 Mathematics Subject Classification. 35Q55.

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Section: (T) = M (U(t)) :=mentioning

confidence: 99%

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Saanouni

2016

Journal of Mathematical Physics

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“…Guo and Xu [12] discussed the physical applications of the fractional Schrödinger equation (FSE). Guo et al [13][14][15][16] investigated the existence and well-posedness criterion of the global smooth solution of the FSE, and obtained the conservation properties including the mass and energy. Han et al [17] established the global well-posedness for the Cauchy problem of fractional SBq (FSBq) equations in H s (R), s ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

Liao

Zhang

2019

Numerical Methods Partial

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In this paper, two conservative finite difference schemes for fractional Schrödinger-Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi-implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis. KEYWORDSconservative law, convergence, discrete energy method, mathematical induction method, Schrödinger-Boussinesq equations 1 Numer Methods Partial Differential Eq. 2019;35:1305-1325. wileyonlinelibrary.com/journal/num

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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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Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

Abstract: The well-posedness for the Cauchy problem of the nonlinear fractional Schrödinger equationis considered. The local well-posedness in subcritical space H s with s > n 2 − α is obtained. Moreover, the inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

Cited by 69 publications

References 11 publications

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

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

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