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Superposition Formulas and Evolution Behaviors of Multi-Solutions to the (3+1)-Dimensional Generalized Shallow Water Wave-like Equation

Author

Listed:
  • Sudao Bilige

    (Department of Mathemaitcs, Inner Mongolia University of Technology, Hohhote 010051, China)

  • Leilei Cui

    (Department of Mathemaitcs, Inner Mongolia University of Technology, Hohhote 010051, China)

  • Xiaomin Wang

    (Department of Mathemaitcs, Inner Mongolia University of Technology, Hohhote 010051, China)

Abstract

The superposition formulas of multi-solutions to the (3+1)-dimensional generalized shallow water wave-like Equation (GSWWLE) are proposed. There are arbitrary test functions in the superposition formulas of the mixed solutions and the interaction solutions, and we generalized to the sum of any N terms. By freely selecting the test functions and the positive integer N , we have obtained abundant solutions for the GSWWLE. First, we introduced new mixed solutions between two arbitrary functions and the multi-kink solitons, and the abundant mixed solutions were obtained through symbolic computation. Next, we constructed the multi-localized wave solutions which are the superposition of N-even power functions. Finally, the novel interaction solutions between the multi-localized wave solutions and the multi-arbitrary function solutions for the GSWWLE were obtained. The evolution behaviors of the obtained solutions are shown through 3D, contour and density plots. The received results have immensely enriched the exact solutions of the GSWWLE in the available literature.

Suggested Citation

  • Sudao Bilige & Leilei Cui & Xiaomin Wang, 2023. "Superposition Formulas and Evolution Behaviors of Multi-Solutions to the (3+1)-Dimensional Generalized Shallow Water Wave-like Equation," Mathematics, MDPI, vol. 11(8), pages 1-12, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1966-:d:1129356
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    References listed on IDEAS

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    1. Wen-Xiu Ma, 2022. "Riemann–Hilbert Problems and Soliton Solutions of Type ( λ ∗ , − λ ∗ ) Reduced Nonlocal Integrable mKdV Hierarchies," Mathematics, MDPI, vol. 10(6), pages 1-21, March.
    2. Jingzhu Wu & Xiuzhi Xing & Xianguo Geng, 2015. "Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-9, September.
    3. Yuefeng Zhou & Chuanjian Wang & Xiaoxue Zhang, 2020. "Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation," Mathematics, MDPI, vol. 8(10), pages 1-13, October.
    4. Xiaomin Wang & Sudao Bilige & Jing Pang, 2020. "Rational Solutions and Their Interaction Solutions of the ( )- Dimensional Jimbo-Miwa Equation," Advances in Mathematical Physics, Hindawi, vol. 2020, pages 1-18, April.
    5. Ma, Wen-Xiu, 2021. "N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 270-279.
    6. Hong-Qian Sun & Zuo-Nong Zhu, 2023. "Darboux Transformation and Soliton Solution of the Nonlocal Generalized Sasa–Satsuma Equation," Mathematics, MDPI, vol. 11(4), pages 1-18, February.
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