Analytical soliton solutions of the Kairat-Ⅱ equation using the Kumar–Malik and extended hyperbolic function methods

Abdul Mateen; Ghulam Hussain Tipu; Loredana Ciurdariu; Fengping Yao; Abdul Mateen; Ghulam Hussain Tipu; Loredana Ciurdariu; Fengping Yao

doi:10.3934/math.2025400

AIMS Mathematics

2025, Volume 10, Issue 4: 8721-8752. doi: 10.3934/math.2025400

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Research article

Analytical soliton solutions of the Kairat-Ⅱ equation using the Kumar–Malik and extended hyperbolic function methods

1.
Ministry of Education Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
2.
Department of Mathematics, Shanghai University, No. 99 Shangda Road, Shanghai 200444, China
3.
Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China
4.
Department of Mathematics, Politehnica University of Timișoara, 300006 Timișoara, Romania

Received: 30 January 2025 Revised: 26 March 2025 Accepted: 02 April 2025 Published: 15 April 2025
MSC : 35Q51, 35C08, 35C07, 35A20

The study of optical solitons has advanced significantly due to their stability and diverse applications, particularly in high-speed telecommunications, optical signal processing, and quantum technologies. This paper focuses on the derivation of exact soliton solutions for the nonlinear Kairat-Ⅱ (K-Ⅱ) equation, which models second-order spatiotemporal and group velocity dispersion effects in nonlinear optical systems. By applying the Kumar–Malik method and the extended hyperbolic function method, a comprehensive set of soliton solutions are obtained, capturing the intricate propagation dynamics of solitons in nonlinear media. The Kumar–Malik method yields numerous soliton solutions such as Jacobi elliptic function solution with an elliptic modulus, the trigonometric function solution, the hyper-trigonometric solution, dark solitons, bright solitons, periodic, singular and rational function solutions, etc. The behavior, stability, and evolution of these solitons are further illustrated through two-dimensional (2D), three-dimensional (3D), and contour plots, providing insights into their structural characteristics under various physical conditions. In this article, new soliton solutions in Jacobi elliptic form are derived for the given equation, providing valuable insights into the theoretical framework of solitons in nonlinear optics and presenting potential advancements for soliton-based technologies.
- Kairat-Ⅱ equation,
- Kumar–Malik method,
- extended hyperbolic function method,
- Jacobi elliptic function,
- traveling wave solutions
Citation: Abdul Mateen, Ghulam Hussain Tipu, Loredana Ciurdariu, Fengping Yao. Analytical soliton solutions of the Kairat-Ⅱ equation using the Kumar–Malik and extended hyperbolic function methods[J]. AIMS Mathematics, 2025, 10(4): 8721-8752. doi: 10.3934/math.2025400

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Abstract

The study of optical solitons has advanced significantly due to their stability and diverse applications, particularly in high-speed telecommunications, optical signal processing, and quantum technologies. This paper focuses on the derivation of exact soliton solutions for the nonlinear Kairat-Ⅱ (K-Ⅱ) equation, which models second-order spatiotemporal and group velocity dispersion effects in nonlinear optical systems. By applying the Kumar–Malik method and the extended hyperbolic function method, a comprehensive set of soliton solutions are obtained, capturing the intricate propagation dynamics of solitons in nonlinear media. The Kumar–Malik method yields numerous soliton solutions such as Jacobi elliptic function solution with an elliptic modulus, the trigonometric function solution, the hyper-trigonometric solution, dark solitons, bright solitons, periodic, singular and rational function solutions, etc. The behavior, stability, and evolution of these solitons are further illustrated through two-dimensional (2D), three-dimensional (3D), and contour plots, providing insights into their structural characteristics under various physical conditions. In this article, new soliton solutions in Jacobi elliptic form are derived for the given equation, providing valuable insights into the theoretical framework of solitons in nonlinear optics and presenting potential advancements for soliton-based technologies.

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