# American Institute of Mathematical Sciences

## 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

Received  January 2020 Revised  April 2020 Published  March 2021

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.

Citation: Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021021
##### References:

show all references

##### References:
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) $\alpha = 0.50$
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 $\xi = 1.$
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) $\alpha = 0.50$
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 $\xi = 1.$
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) $\alpha = 0.50$
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 $\xi = 1.$
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) $\alpha = 0.50$
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 $\xi = 1.$
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) $\alpha = 0.50$
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 $\xi = 1.$
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) $\alpha = 0.50$
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 $\xi = 1.$
 [1] Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021021 [2] Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013 [3] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376 [4] Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024 [5] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038 [6] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [7] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [8] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [9] Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216 [10] Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014 [11] Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115 [12] Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392 [13] Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100 [14] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [15] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [16] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [17] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 [18] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [19] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [20] Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

2019 Impact Factor: 1.233

## Tools

Article outline

Figures and Tables