Volume 10, Issue 2
A Jacobi-Galerkin Spectral Method for Computing the Ground and First Excited States of Nonlinear Fractional Schrödinger Equation

Ying Ma & Lizhen Chen

East Asian J. Appl. Math., 10 (2020), pp. 274-294.

Published online: 2020-04

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  • Abstract

The behaviour of the ground and first excited states of the nonlinear fractional Schrödinger equation is studied by an approximation method. In order to determine the nonlinear term of the problem under consideration, a normalised fractional gradient flow is introduced and the decay of a modified energy is established. The problem is then discretised by a semi-implicit Euler method in time and Jacobi-Galerkin spectral method in space. One- and two-dimensional numerical examples show that the strong nonlocal interactions lead to a large scattering of particles. Moreover, numerical simulations confirm the fundamental gap conjecture and show that for small interactions the ground and first excited states are more peaked and narrower.

  • Keywords

Fractional Schrödinger equation, semi-implicit Euler method, ground state, first excited state.

  • AMS Subject Headings

65N25, 65N30, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yingma@csrc.ac.cn (Ying Ma)

lzchen@csc.ac.cn (Lizhen Chen)

  • BibTex
  • RIS
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@Article{EAJAM-10-274, author = {Ma , Ying and Chen , Lizhen }, title = {A Jacobi-Galerkin Spectral Method for Computing the Ground and First Excited States of Nonlinear Fractional Schrödinger Equation}, journal = {East Asian Journal on Applied Mathematics}, year = {2020}, volume = {10}, number = {2}, pages = {274--294}, abstract = {

The behaviour of the ground and first excited states of the nonlinear fractional Schrödinger equation is studied by an approximation method. In order to determine the nonlinear term of the problem under consideration, a normalised fractional gradient flow is introduced and the decay of a modified energy is established. The problem is then discretised by a semi-implicit Euler method in time and Jacobi-Galerkin spectral method in space. One- and two-dimensional numerical examples show that the strong nonlocal interactions lead to a large scattering of particles. Moreover, numerical simulations confirm the fundamental gap conjecture and show that for small interactions the ground and first excited states are more peaked and narrower.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.140319.180719}, url = {http://global-sci.org/intro/article_detail/eajam/16138.html} }
TY - JOUR T1 - A Jacobi-Galerkin Spectral Method for Computing the Ground and First Excited States of Nonlinear Fractional Schrödinger Equation AU - Ma , Ying AU - Chen , Lizhen JO - East Asian Journal on Applied Mathematics VL - 2 SP - 274 EP - 294 PY - 2020 DA - 2020/04 SN - 10 DO - http://dor.org/10.4208/eajam.140319.180719 UR - https://global-sci.org/intro/article_detail/eajam/16138.html KW - Fractional Schrödinger equation, semi-implicit Euler method, ground state, first excited state. AB -

The behaviour of the ground and first excited states of the nonlinear fractional Schrödinger equation is studied by an approximation method. In order to determine the nonlinear term of the problem under consideration, a normalised fractional gradient flow is introduced and the decay of a modified energy is established. The problem is then discretised by a semi-implicit Euler method in time and Jacobi-Galerkin spectral method in space. One- and two-dimensional numerical examples show that the strong nonlocal interactions lead to a large scattering of particles. Moreover, numerical simulations confirm the fundamental gap conjecture and show that for small interactions the ground and first excited states are more peaked and narrower.

Ying Ma & Lizhen Chen. (2020). A Jacobi-Galerkin Spectral Method for Computing the Ground and First Excited States of Nonlinear Fractional Schrödinger Equation. East Asian Journal on Applied Mathematics. 10 (2). 274-294. doi:10.4208/eajam.140319.180719
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