New solutions for the unstable nonlinear Schrödinger equation arising in natural science

Mahmoud A. E. Abdelrahman; Sherif I. Ammar; Kholod M. Abualnaja; Mustafa Inc; Mahmoud A. E. Abdelrahman; Sherif I. Ammar; Kholod M. Abualnaja; Mustafa Inc

doi:10.3934/math.2020126

AIMS Mathematics

2020, Volume 5, Issue 3: 1893-1912. doi: 10.3934/math.2020126

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New solutions for the unstable nonlinear Schrödinger equation arising in natural science

1.
Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
3.
Department of Mathematics, Faculty of Science, Menofia University, Menofia, Egypt
4.
Department of Mathematics, Umm Al-Qura University, P.O. Box 14949, Makkah, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Firat University, 23119 Elazig/Turkey

Received: 08 November 2019 Accepted: 02 January 2020 Published: 18 February 2020
MSC : 35A20, 35A99, 83C15, 65Z05

In this work, three mathematical methods are applied, namely, the exp(-φ(ξ))-expansion method, the sine-cosine technique and the Riccati-Bernoulli sub-ODE method for constructing many new exact solutions of the unstable nonlinear Schrödinger equation. The exact solutions are obtained in the form of rational, exponential, trigonometric, hyperbolic functions. These solutions may be so important significance for the explanation of some practical physical problems. The computational work and obtained results show that the presented methods are simple, efficient, straightforward and powerful. Moreover, the presented methods can be employed to many other types of nonlinear partial differential equations arising in mathematics, mathematical physics and other areas of natural sciences. Some solutions are simulated for some particular choices of parameters.
- Schrödinger equation,
- exp(-φ(ξ))-expansion method,
- sine-cosine method,
- Riccati-Bernoulli sub-ODE method,
- new exact solutions
Citation: Mahmoud A. E. Abdelrahman, Sherif I. Ammar, Kholod M. Abualnaja, Mustafa Inc. New solutions for the unstable nonlinear Schrödinger equation arising in natural science[J]. AIMS Mathematics, 2020, 5(3): 1893-1912. doi: 10.3934/math.2020126

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Abstract

In this work, three mathematical methods are applied, namely, the exp(-φ(ξ))-expansion method, the sine-cosine technique and the Riccati-Bernoulli sub-ODE method for constructing many new exact solutions of the unstable nonlinear Schrödinger equation. The exact solutions are obtained in the form of rational, exponential, trigonometric, hyperbolic functions. These solutions may be so important significance for the explanation of some practical physical problems. The computational work and obtained results show that the presented methods are simple, efficient, straightforward and powerful. Moreover, the presented methods can be employed to many other types of nonlinear partial differential equations arising in mathematics, mathematical physics and other areas of natural sciences. Some solutions are simulated for some particular choices of parameters.

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