Study of the N-Soliton Solutions for the (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

Ai Jing; Wuming Li

doi:https://doi.org/10.14445/22315373/IJMTT-V72I8P107

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 8 | Year 2026 | Article Id. IJMTT-V72I8P107 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I8P107

Study of the N-Soliton Solutions for the (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

Ai Jing, Wuming Li

Received	Revised	Accepted	Published
27 Jun 2026	28 Jul 2026	18 Aug 2026	30 Aug 2026

Citation :

Ai Jing, Wuming Li, "Study of the N-Soliton Solutions for the (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 8, pp. 54-75, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I8P107

Abstract

This paper investigates the N-soliton solutions of the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation. By virtue of the Hirota bilinear method, the bilinear form of the equation is derived, and the analytical expression for its N-soliton solutions is constructed. When 𝑁≥2 , with reasonable parameter constraints and appropriate parameter selections, the multi-soliton solutions can degenerate into other types of solitons and exhibit the nonlinear interaction characteristics between different solitons. For instance, two-line solitons can transform into a periodic soliton, and a two-kink soliton can simplify to a kink breather soliton. Three line solitons can present the interaction between a line soliton and a periodic soliton. In contrast, a three-kink soliton can exhibit the interaction between a single-kink soliton and a kink breather soliton. Meanwhile, the four-soliton solution can effectively describe the collision and evolution processes between two line solitons and one periodic soliton, as well as the interaction between two periodic solitons. In addition, applying the long-wave limit method yields lump soliton solutions, and the interaction mechanism between lump solitons and line solitons is examined. Numerical simulations are conducted for the derived soliton structures, and their spatial evolution diagrams are presented. Furthermore, the dynamic characteristics and propagation rules of these solitons are systematically investigated. This finding enriches the solutions to this equation, broaden research on soliton categories and interaction mechanisms, and offer theoretical references for analyzing relevant nonlinear physical models.

Keywords

(2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Line Soliton, Kink Soliton, Periodic Soliton, Lump Soliton.

References

[1] Philip G. Drazin, and Robin Stanley Johnson, Solitons: An Introduction, 2^nd ed., Cambridge University Press, 1989.
[Publisher Link]

[2] Shi-Jun Liao, “A Kind of Approximate Solution Technique which does not Depend Upon Small Parameters—II. An Application in Fluid Mechanics,” International Journal of Non-Linear Mechanics, vol. 32, no. 5, pp. 815-822, 1997.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Chong-Dong Cheng et al., “Pfaffian, Breather, and Hybrid Solutions for a (2+1)-Dimensional Generalized Nonlinear System in Fluid Mechanics and Plasma Physics,” Physics of Fluids, vol. 34, no. 11, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[4] Rashid Ali, Zhao Zhang, and Hijaz Ahmad, “Exploring Soliton Solutions in Nonlinear Spatiotemporal Fractional Quantum Mechanics Equations: An Analytical Study,” Optical and Quantum Electronics, vol. 56, no. 5, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Waqas Ali Faridi et al., “The Fractional Soliton Solutions of Dynamical System Arising in Plasma Physics: The Comparative Analysis,” Alexandria Engineering Journal, vol. 95, pp. 247-261, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Sachin Kumar, Brij Mohan, and Raj Kumar, “Lump, Soliton, and Interaction Solutions to a Generalized Two-Mode Higher-Order Nonlinear Evolution Equation in Plasma Physics,” Nonlinear Dynamics, vol. 110, no. 1, pp. 693-704, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Bart Lannoo et al., “Radio-Over-Fiber-based Solution to Provide Broadband Internet Access to Train Passengers [Topics in Optical Communications],” IEEE Communications Magazine, vol. 45, no. 2, pp. 56-62, 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Wen-Xiu Ma et al., “Hirota Bilinear Equations with Linear Subspaces of Solutions,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7174-7183, 2012.
[CrossRef] [Google Scholar] [Publisher Link]

[9] Marwan Alquran, and Rahaf Alhami, “Analysis of Lumps, Single-Stripe, Breather-Wave, and Two-Wave Solutions to the Generalized Perturbed-KdV Equation by Means of Hirota’s Bilinear Method,” Nonlinear Dynamics, vol. 109, no. 3, pp. 1985-1992, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Nazia Batool et al., “Hirota Bilinear Method and Multi-Soliton Interaction of Electrostatic Waves Driven by Cubic Nonlinearity in Pair-Ion–Electron Plasmas,” Physics of Fluids, vol. 35, no. 3, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Yueyang Feng, and Sudao Bilige, “Multiple Rogue Wave Solutions of (2+1)-Dimensional YTSF Equation via Hirota Bilinear Method,” Wave in Random Complex, vol. 31, no. 1, pp. 94-110, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[12] Deniz Bilman, and Peter D. Miller, “A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation,” Communications on Pure and Applied Mathematics, vol. 72, no. 8, pp. 1722-1805, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Hai-Qiang Zhang et al., “Optical Soliton Solutions for Two Coupled Nonlinear Schrödinger Systems via Darboux Transformation,” Physica Scripta, vol. 76, no. 5, pp. 452-460, 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[14] Li-Li Wen, and Hai-Qiang Zhang, “Darboux Transformation and Soliton Solutions of the (2+1)-Dimensional Derivative Nonlinear Schrödinger Hierarchy,” Nonlinear Dynamics, vol. 84, no. 2, pp. 863-873, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[15] Xing-Yu Luo, and Yong Chen, “Darboux Transformation and N-Soliton Solution for Extended form of Modified Kadomtsev—Petviashvili Equation with Variable-Coefficient,” Communications in Theoretical Physics, vol. 66, no. 2, pp. 179-188, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[16] Vladislav Gavrilovich Bagrov, and Boris Fedorovich Samsonov, “Darboux Transformation, Factorization, and Supersymmetry in One-Dimensional Quantum Mechanics,” Theoretical and Mathematical Physics, vol. 104, no. 2, pp. 1051-1060, 1995.
[CrossRef] [Google Scholar] [Publisher Link]

[17] Ying Shi et al., “Abundant Analytic Solutions of the Stochastic KdV Equation with Time-Dependent Additive White Gaussian Noise via Dkwarboux Transformation Method,” Nonlinear Dynamics, vol. 111, no. 3, pp. 2651- 2661, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Xin-Yi Gao, “In an Ocean or a River: Bilinear Auto-Bäcklund Transformations and Similarity Reductions on an Extended Time-Dependent (3+1)-Dimensional Shallow Water Wave Equation,” China Ocean Engineering, vol. 39, no. 1 pp. 160-165, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[19] Yi Wang, and Xing Lü, “Bäcklund Transformation and Interaction Solutions of a Generalized Kadomtsev–Petviashvili Equation with Variable Coefficients,” Chinese Journal of Physics, vol. 89, pp. 37-45, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[20] Tian-Yu Zhou et al., “Auto-Bäcklund Transformations and Soliton Solutions on the Nonzero Background for a (3+1)-Dimensional Korteweg-De Vries-Calogero-Bogoyavlenskii-Schif Equation in a Fluid,” Nonlinear Dynamics, vol. 111, no. 9, pp. 8647-8658, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[21] Tian-Yu Zhou et al., “Painlevé Analysis, Auto-Bäcklund Transformation and Analytic Solutions of a (2+1)- Dimensional Generalized Burgers System with the Variable Coefficients in a Fluid,” Nonlinear Dynamics, vol. 108, no. 3, pp. 2417-2428, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[22] Yue Kai, Jialiang Ji, and Zhixiang Yin, “Study of the Generalization of Regularized Long-Wave Equation,” Nonlinear Dynamics, vol. 107, no. 3, pp. 2745-2752, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[23] Xin Wu, Yong Chen, and Xue-Wei Yan, “General Soliton, Line Breather and (Semi-) Rational Solutions for the Nonlocal Long-Wave-Short-Wave Resonance Interaction Equation,” Nonlinear Dynamics, vol. 112, no. 1, pp. 661-679, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[24] Luis A. Cisneros-Ake, and José F. Solano Peláez, “Bright and Dark Solitons in the Unidirectional Long Wave Limit for the Energy Transfer on Anharmonic Crystal Lattices,” Physica D: Nonlinear Phenomena, vol. 346, pp. 20-27, 2017.
[CrossRef] [Google Scholar] [Publisher Link]

[25] Chi-Kun Lin, and Yau-Shu Wong, “Zero-Dispersion Limit of the Short-Wave–Long-Wave Interaction Equations,” Journal of Differential Equations, vol. 228, no. 1, pp. 87-110, 2006.
[CrossRef] [Google Scholar] [Publisher Link]

[26] Kang-Jia Wang, “Soliton Molecules, Y-Type Soliton and Complex Multiple Soliton Solutions to the Extended (3+1)-Dimensional Jimbo-Miwa Equation,” Physica Scripta, vol. 99, no. 1, pp. 2014.
[CrossRef] [Google Scholar] [Publisher Link]

[27] Sachin Kumar, and Shubham Kumar Dhiman, “Exploring Cone-Shaped Solitons, Breather, and Lump-Forms Solutions using the Lie Symmetry Method and Unified Approach to a Coupled Breaking Soliton Model,” Physica Scripta, vol. 99, no. 2, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[28] Xinying Yan, Jinzhou Liu, and Xiangpeng Xin, “Soliton Solutions and Lump-Type Solutions to the (2+1)- Dimensional Kadomtsev-Petviashvili Equation with Variable Coefficient,” Physics Letters A, vol. 457, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[29] Hongcai Ma et al., “Lump Solution, Breather Soliton and More Soliton Solutions for a (2+1)-Dimensional Generalized Benjamin–Ono Equation,” Qualitative Theory of Dynamical Systems, vol. 22, no. 2, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[30] Hongcai Ma, Shupan Yue, and Aiping Deng, “Resonance Y-Shape Solitons and Mixed Solutions for a (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Fluid Mechanics,” Nonlinear Dynamics, vol. 108, no. 1, pp. 505-519, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[31] Wei-Qi Peng et al., “Characteristics of the Solitary Waves and Lump Waves with Interaction Phenomena in a (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation,” Nonlinear Dynamics, vol. 93, no. 4, pp. 1841-1851, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[32] Gao-Fu Deng et al., “Solitons and Periodic Waves for the (2+1)-Dimensional Generalized Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Fluid Mechanics,” Nonlinear Dynamics, vol. 99, no. 2, pp. 1039-1052, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[33] Li-Jun Xu et al., “The Multi‑Soliton Solutions for the (2+1)‑Dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada Equation,” Mathematics, vol. 13, no. 2, pp. 1-13, 2025.
[CrossRef] [Google Scholar] [Publisher Link]