On the orbital stability of fractional Schrödinger equations

Cho, Yonggeun; Hajaiej, Hichem; Hwang, Gyeongha; Ozawa, Tohru

doi:10.3934/cpaa.2014.13.1267

Cited by 42 publications

(25 citation statements)

References 19 publications

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“…The existence of a non-trivial solution of Eq. (1.5) has been studied in [20,36], and the stability of related standing waves has been obtained in [9,14,35]. In [36], the second author of this paper obtained a sharp Gagliardo-Nirenberg inequality, which reveals the variational characteristic of the ground-state solutions for Eq.…”

Section: Introductionmentioning

confidence: 87%

Sharp threshold of blow-up and scattering for the fractional Hartree equation

Guo

Zhu

2018

Journal of Differential Equations

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We consider the fractional Hartree equation in the L 2 -supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: Ifs−sc sc Q 2Ḣ s , the solution u(t) blows up in finite time. This condition is sharp in the sense that the solitary wave solution e it Q(x) is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation. MSC: 35Q40, 35Q55, 47J30

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Section: Introductionmentioning

confidence: 87%

Sharp threshold of blow-up and scattering for the fractional Hartree equation

Guo

Zhu

2018

Journal of Differential Equations

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“…Although a general existence theorem for blow-up solutions of this problem has remained an open problem, it has been strongly supported by numerical evidence [20]. The orbitally stability of standing waves for other kinds of fractional Schrödinger equations has been studied in [12,13,4,40].…”

Section: Introductionmentioning

confidence: 99%

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Feng¹

2018

Communications on Pure &Amp; Applied Analysis

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This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities2s . Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blowup and global existence in the case 0 < p 1 < 2s N and p 2 = 2s N . Finally, we investigate the dynamical properties of blow-up solutions, including L 2 -concentration, blow-up rate and limiting profile. Keywords: The fractional Schrödinger equation; Blow-up solutions; Combined powertype nonlinearities; Sharp thresholds; The dynamical behavior where 0 < s < 1, f (u) = |u| 2p u. The fractional differential operator (−∆) s is defined by (−∆) s u = F −1 [|ξ| 2s F(u)], where F and F −1 are the Fourier transform and inverse Fourier transform, respectively.

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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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On the orbital stability of fractional Schrödinger equations

Abstract: We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.

Cited by 42 publications

References 19 publications

Sharp threshold of blow-up and scattering for the fractional Hartree equation

Sharp threshold of blow-up and scattering for the fractional Hartree equation

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

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