On a fractional Schrödinger equation in the presence of harmonic potential

Zhiyan Ding; Hichem Hajaiej; Zhiyan Ding; Hichem Hajaiej

doi:10.3934/era.2021047

Electronic Research Archive

2021, Volume 29, Issue 5: 3449-3469. doi: 10.3934/era.2021047

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On a fractional Schrödinger equation in the presence of harmonic potential

Zhiyan Ding ^{1
,},
Hichem Hajaiej ^{2
,
,}

1.
Mathematics Department, University of Wisconsin-Madison, Madison, WI 53706, USA
2.
Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA

Received: 01 November 2020 Revised: 01 March 2021 Published: 24 June 2021
Primary: 35Q55; Secondary: 47J30, 35J60

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations [11,18]. Our results explain the effect of each term of the Schrödinger equation: the fractional power, the power of the nonlinearity, and the harmonic potential.
- Schrödinger equation,
- fractional Laplacian,
- harmonic potential,
- standing waves,
- variational methods
Citation: Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential[J]. Electronic Research Archive, 2021, 29(5): 3449-3469. doi: 10.3934/era.2021047

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Abstract

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations [11,18]. Our results explain the effect of each term of the Schrödinger equation: the fractional power, the power of the nonlinearity, and the harmonic potential.

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