Bifurcation and solitary wave solutions of a time-dependent paraxial equation using improved modified extended tanh function method

Arooma Zainab; Muhammad Abbas; Yagoub A. S. Arko; Alina Alb Lupas; Tahir Nazir; Muhammad Zain Yousaf; Arooma Zainab; Muhammad Abbas; Yagoub A. S. Arko; Alina Alb Lupas; Tahir Nazir; Muhammad Zain Yousaf

doi:10.3934/math.20251001

AIMS Mathematics

2025, Volume 10, Issue 9: 22471-22496. doi: 10.3934/math.20251001

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Bifurcation and solitary wave solutions of a time-dependent paraxial equation using improved modified extended tanh function method

1.
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
2.
Department of Mathematics, Turabah University College, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
3.
Department of Mathematics and Computer Science, University of Oradea, Oradea 410087, Romania

Received: 28 July 2025 Revised: 07 September 2025 Accepted: 15 September 2025 Published: 28 September 2025
MSC : 34G20, 35A20, 35A22, 35R11

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.
- improved modified extended tanh function method,
- dimensionless time-dependent paraxial equation,
- bifurcation,
- soliton solutions,
- stability
Citation: Arooma Zainab, Muhammad Abbas, Yagoub A. S. Arko, Alina Alb Lupas, Tahir Nazir, Muhammad Zain Yousaf. Bifurcation and solitary wave solutions of a time-dependent paraxial equation using improved modified extended tanh function method[J]. AIMS Mathematics, 2025, 10(9): 22471-22496. doi: 10.3934/math.20251001

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Abstract

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.

References

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