New soliton wave structure and modulation instability analysis for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities

Abeer S. Khalifa; Hamdy M. Ahmed; Niveen M. Badra; Wafaa B. Rabie; Farah M. Al-Askar; Wael W. Mohammed; Abeer S. Khalifa; Hamdy M. Ahmed; Niveen M. Badra; Wafaa B. Rabie; Farah M. Al-Askar; Wael W. Mohammed

doi:10.3934/math.20241278

AIMS Mathematics

2024, Volume 9, Issue 9: 26166-26181. doi: 10.3934/math.20241278

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New soliton wave structure and modulation instability analysis for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities

1.
Department of Physics and Engineering Mathematics, Faculty of Engineering, Ain Shams University, Abbassia, Cairo, Egypt
2.
Department of Mathematics, Faculty of Basic Sciences, The German University in Cairo (GUC), Cairo, Egypt
3.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt
4.
Department of Engineering Mathematics and Physics, Higher Institute of Engineering and Technology, Tanta, Egypt
5.
Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
6.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
7.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 21 July 2024 Revised: 14 August 2024 Accepted: 30 August 2024 Published: 10 September 2024
MSC : 35B35, 35C07, 35C08, 35C09

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.
- nonlinear Schrödinger equation,
- modified extended direct algebraic method,
- solitons,
- modulation instability
Citation: Abeer S. Khalifa, Hamdy M. Ahmed, Niveen M. Badra, Wafaa B. Rabie, Farah M. Al-Askar, Wael W. Mohammed. New soliton wave structure and modulation instability analysis for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities[J]. AIMS Mathematics, 2024, 9(9): 26166-26181. doi: 10.3934/math.20241278

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Abstract

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.

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