On fractional Schrödinger equation in -dimensional fractional space

Abstract：We report solutions for solitons of the "accessible" type in globally nonlocal nonlinear media of fractional dimension (FD), viz., for self-trapped modes in the space of effective dimension 3 2   D with harmonic-oscillator potential whose strength is proportional to the total power of the wave field. The solutions are categorized by a combination of radial, orbital and azimuthal quantum numbers ) , , ( m l n . They feature coaxial sets of vortical and necklace-shaped rings of different orders, and can be exactly written in terms of special functions that include Gegenbauer polynomials, associated Laguerre polynomials, and associated Legendre functions. The validity of these solutions is verified by direct simulation. The model can be realized in various physical settings emulated by FD spaces; in particular, it applies to excitons trapped in quantum wells.

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“…, admitting solutions in terms of Gegenbauer polynomials, [27][28][29] and associated Legendre functions,…”

Section: The Modelmentioning

confidence: 99%

Spatiotemporal accessible solitons in fractional dimensions

et al. 2016

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“…Using v x (r) as an effective scalar field ϕ ef f (r) = v x (r), we can apply equations (22), (21) and (24) where (48). Equation (48) with α r = α x = 1 gives (35).…”

Section: Fractal Fluid With α R =mentioning

confidence: 99%

Flow of fractal fluid in pipes: Non-integer dimensional space approach

Tarasov

2014

Chaos, Solitons & Fractals

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Using a generalization of vector calculus for the case of non-integer dimensional space we consider a Poiseuille flow of an incompressible viscous fractal fluid in the pipe. Fractal fluid is described as a continuum in non-integer dimensional space. A generalization of the Navier-Stokes equations for non-integer dimensional space, its solution for steady flow of fractal fluid in a pipe and corresponding fractal fluid discharge are suggested.

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“…Coarse-grained and thus fractal spaces are basic in El Naschie's work [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Fractals are basic in Nottale's work [52][53][54][55], and transparent in Ord's paper [57].…”

Section: Coarse-grained Space-time In Physicsmentioning

confidence: 99%

Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space

Jumarie

2009

J. Appl. Math. Comput.

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Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modelled as a space in which the generic increment is not dx, but rather (dx) α , 0 < α < 1. The purpose of the article is to analyze the non-linearity so induced by this coarse-graining effect. This approach via (dx) α leads us to the use of fractional analysis which so provides models in the form of nonlinear differential equations of fractional order. The purpose of the paper is to examine how the coarse-graining effect affects the equilibrium position of dynamical systems, and to this end, one uses the linearization technique on the one hand, and the Lyapunov function approach on the other hand in the framework of fractional analysis. It is shown that in some instances, the coarsegraining phenomenon can convert the initial stable system into an unstable system.

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On fractional Schrödinger equation in -dimensional fractional space

Cited by 55 publications

References 10 publications

Spatiotemporal accessible solitons in fractional dimensions

Spatiotemporal accessible solitons in fractional dimensions

Flow of fractal fluid in pipes: Non-integer dimensional space approach

Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space

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