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Volume 13
Issue 16
10.3390/math13162686
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes

by
Fadwa Althrwi
1,
Aisha S. H. Farhat
1,
A. A. AlQarni
2,*,
H. O. Bakodah
1 and
A. A. Alshaery
1
1
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23445, Saudi Arabia
2
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(16), 2686; https://doi.org/10.3390/math13162686
Submission received: 6 July 2025 / Revised: 11 August 2025 / Accepted: 12 August 2025 / Published: 20 August 2025
Versions Notes

Abstract

Analytical solutions for the complex-valued nonlinear Gerdjikov–Ivanov (GI) equation have been studied extensively using integrability-based methods. In contrast, numerical and semi-analytical exploration remains relatively underdeveloped. Thus, the present study deploys both the traditional Adomian decomposition method (ADM) and its improved version (IADM) to explore the computational relevance of the GI equation to shock waves against a benchmark exact soliton solution. The findings indicate that both methods are effective in addressing the GI equation, with the improved method demonstrating an enhancement in the stability of the convergence under specific conditions. This work offers the first systematic semi-analytic and numerical evaluation of the GI equation, introducing practical implementation guidelines.
Keywords: Gerdjikov–Ivanov equation; nonlinear Schrödinger’s equation; shock waves; decomposition methods; nonlinear optics; optical communications Gerdjikov–Ivanov equation; nonlinear Schrödinger’s equation; shock waves; decomposition methods; nonlinear optics; optical communications

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MDPI and ACS Style

Althrwi, F.; Farhat, A.S.H.; AlQarni, A.A.; Bakodah, H.O.; Alshaery, A.A. Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes. Mathematics 2025, 13, 2686. https://doi.org/10.3390/math13162686

AMA Style

Althrwi F, Farhat ASH, AlQarni AA, Bakodah HO, Alshaery AA. Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes. Mathematics. 2025; 13(16):2686. https://doi.org/10.3390/math13162686

Chicago/Turabian Style

Althrwi, Fadwa, Aisha S. H. Farhat, A. A. AlQarni, H. O. Bakodah, and A. A. Alshaery. 2025. "Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes" Mathematics 13, no. 16: 2686. https://doi.org/10.3390/math13162686

APA Style

Althrwi, F., Farhat, A. S. H., AlQarni, A. A., Bakodah, H. O., & Alshaery, A. A. (2025). Shock Waves of the Gerdjikov–Ivanov Equation Using the Adomian Decomposition Schemes. Mathematics, 13(16), 2686. https://doi.org/10.3390/math13162686

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