Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator

Amit Goswami; Sushila Rathore; Jagdev Singh; Devendra Kumar

doi:10.3934/dcdss.2021021

Article Contents

2021, Volume 14, Issue 10: 3589-3610. Doi: 10.3934/dcdss.2021021

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Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator

1.
Department of Physics, Jagan Nath University, Jaipur-303901, Rajasthan, India
2.
Department of Physics, Vivekananda Global University, Jaipur-303012, Rajasthan, India
3.
Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India
4.
Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India

^* Corresponding author: Sushila
^* Corresponding author: Sushila

Received: December 31, 2019

Revised: March 31, 2020

Early access: March 2021

Published: October 2021

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Abstract

In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motion of quantum oscillator, lattice vibration, propagation of electromagnetic waves, fluid flow, etc. The main objective of this study is to show the effectiveness of HPSTM, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results reveal that proposed scheme is a powerful tool for study large class of problems. This study shows that the results obtained by the HPSTM are accurate and effective for analysis the nonlinear behaviour of complex systems and efficient over other available analytical schemes.

Keywords:

Mathematics Subject Classification: Primary: 26A33, 35Q41, 35R11.

Citation:

Full Text(HTML)

Figure 1. Surface shows the imaginary part of the 1-D wave function $ \psi \left(\xi ,\eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 2. Comparison of profile of imaginary part of $ \psi \left(\xi ,\eta \right) $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $(b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

Figure 3. Surface shows the real part of the 1-D wave function $ \psi \left(\xi ,\eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 4. Comparison of profile of real part of $ \psi \left(\xi ,\eta \right) $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $(b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

Figure 5. Surface shows the imaginary part of the 2-D wave function $ \psi \left(\xi ,\zeta, \eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 6. Comparison of profile of imaginary part of $ \psi \left(\xi ,\zeta,\eta \right) $ for $ \zeta = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $(b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

Figure 7. Surface shows the real part of the 2-D wave function $ \psi \left(\xi ,\zeta, \eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 8. Comparison of profile of real part of $ \psi \left(\xi ,\zeta,\eta \right) $ for $ \zeta = 1 $at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $(b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

Figure 9. Surface shows the imaginary part of the 3-D wave function $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 10. Comparison of profile of imaginary part of $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ for $ \zeta = \chi = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $ (b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

Figure 11. Surface shows the real part of the 3-D wave function $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ at (a) $ \alpha = 1 $, (b) $ \alpha = 0.75 $ and (c) $ \alpha = 0.50 $

Download: Full-size image PowerPoint slide

Figure 12. Comparison of profile of real part of $ \psi \left(\xi ,\zeta,\chi, \eta \right) $ for $ \zeta = \chi = 1 $ at different values of $ \alpha $ (a) $ -5\le \xi \le 5 $ and $ \eta = 0.01 $(b) $ 0\le \eta \le 1 $ and $ \xi = 1. $

Download: Full-size image PowerPoint slide

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