Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves

Jibrin Sale Yusuf

doi:doi:10.11648/j.ijamtp.20251101.11

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Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves

Jibrin Sale Yusuf^*

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)

Received: 30 October 2024 Accepted: 10 December 2024 Published: 20 January 2025

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Abstract

This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)
DOI	10.11648/j.ijamtp.20251101.11
Page(s)	1-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Optical Soliton, Unstable NLSE, Instability, Rogue Waves

References

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[2]	M. Arshad, Aly R. S. Dianchen L. Wang, Modulation instability analysis of modify unstable nonlinear schrodinger dynamical equation and its optical soliton solutions. Results in Physics 7 (2017) 4153-4161.
[3]	A. Mahmoud, E. Abdelrahman, S. I. Ammar, K. M. Abualnaja and M. Inc, New solutions for the unstable nonlinear Schrodinger equation arising in natural science. AIMS Mathematics, 5(3): 1893-1912. https://doi.org/10.3934/math.2020126
[4]	Sarwar, A.; Arshad, M.; Farman, M.; Akgül, A.; Ahmed, I.; Bairam, M.; Rezapour, S.; De la Sen, M. Construction of Novel Bright-Dark Solitons and Breather Waves of Unstable Nonlinear Schrödinger Equations with Applications. Symmetry 2023, 15, 99. https://doi.org/10.3390/sym15010099
[5]	Yue, L.; Lu, D.; Arshad, M. Xu, X. New exact traveling wave solutions of the unstable nonlinear Schrödinger equations and their applications. Optik 2021, 226, 165386.
[6]	Aliyu AI, Inc M, Yusuf A, Baleanu D and Bayram M (2019) Dark-Bright Optical Soliton and Conserved Vectors to the Biswas-Arshed Equation With Third-Order Dispersions in the Absence of Self-Phase Modulation. Front. Phys. 7: 28. https://doi.org/10.3389/fphy.2019.00028
[7]	He, J. S.; Charalampidis, E. G.; Kevrekidis, P. G.; Frantzeskakis, D. J. Rogue waves in nonlinear Schrödinger models with variable coefficients: Application to Bose-Einstein condensates. Phys. Lett. A 2014, 378, 577-583.
[8]	L. Dianchen, A. Seadawy b, M. Arshad, Applications of extended simple equation method on unstable nonlinear Schrödinger equations, Optik. Volume 140, July 2017, Pages 136-144.
[9]	E. Tala-Tebue, Z. I. Djoufack, E. Fendzi-Donfack, A. Kenfack-Jiotsa, T. C. Kofané, Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method, Optik Volume 127, Issue 23, December 2016, Pages 11124-11130.
[10]	Tian Shou-Fu, Zou Li, Ding Qi, Zhang Hong-Qing. Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrdinger equation with time-dependent nonlinearity. Commun Nonlinear Sci Numer Simul 2012; 17(8): 3247-57.
[11]	K. Hosseini, D. Kumar, M. Kaplan and E. Y Bejarbaneh, New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrödinger Equations Commun. Theor. Phys. 68, 761. https://doi.org/10.1088/0253-6102/68/6/761
[12]	Aliyu, A. I.; Yusuf, J. S.; Nauman, M. M.; Ozsahin, D. U.; Agaie, B. G.; Zaini, J. H.; Umar, H. Lie Symmetry Analysis and Explicit Solutions of the Estevez -Mansfield-Clarkson Equation. Mdpi, Journal of Symmetry 2024, 1, 0. https://www.mdpi.com/journal/symmetry.doi.org/10.3390/sym1010000
[13]	GaladimaB.A,YusufJ.S,AliyuA.I,A.AWachinandS. U Zuwaira: Optical Soliton Solutions of Burgers-Fisher and Burgers-Huxley Equations. KASU JOURNAL OF MATHEMATICAL SCIENCES (KJMS) VOL. 5, ISSUE 1, JUNE 2024. https://doi.org/10.5281/zenodo.12627471
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APA Style

Yusuf, J. S. (2025). Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 1-18. https://doi.org/10.11648/j.ijamtp.20251101.11

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ACS Style

Yusuf, J. S. Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 1-18. doi: 10.11648/j.ijamtp.20251101.11

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AMA Style

Yusuf JS. Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. Int J Appl Math Theor Phys. 2025;11(1):1-18. doi: 10.11648/j.ijamtp.20251101.11

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@article{10.11648/j.ijamtp.20251101.11,
  author = {Jibrin Sale Yusuf},
  title = {Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {1-18},
  doi = {10.11648/j.ijamtp.20251101.11},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.11},
  abstract = {This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.},
 year = {2025}
}

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TY  - JOUR
T1  - Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves
AU  - Jibrin Sale Yusuf
Y1  - 2025/01/20
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.11
DO  - 10.11648/j.ijamtp.20251101.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 1
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.11
AB  - This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.
VL  - 11
IS  - 1
ER  -

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Author Information

Jibrin Sale Yusuf

Department of Mathematics, Federal University, Dutse, Nigeria

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APA Style

Yusuf, J. S. (2025). Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 1-18. https://doi.org/10.11648/j.ijamtp.20251101.11

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ACS Style

Yusuf, J. S. Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 1-18. doi: 10.11648/j.ijamtp.20251101.11

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AMA Style

Yusuf JS. Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves. Int J Appl Math Theor Phys. 2025;11(1):1-18. doi: 10.11648/j.ijamtp.20251101.11

Copy | Download

@article{10.11648/j.ijamtp.20251101.11,
  author = {Jibrin Sale Yusuf},
  title = {Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {1-18},
  doi = {10.11648/j.ijamtp.20251101.11},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.11},
  abstract = {This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves
AU  - Jibrin Sale Yusuf
Y1  - 2025/01/20
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.11
DO  - 10.11648/j.ijamtp.20251101.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 1
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.11
AB  - This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.
VL  - 11
IS  - 1
ER  -

Copy | Download