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N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation

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  • Ma, Wen-Xiu

Abstract

Within the Hirota bilinear formulation, we construct N-soliton solutions and analyze the Hirota N-soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N-soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation, as specific examples.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:270-279
    DOI: 10.1016/j.matcom.2021.05.020
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    Citations

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    Cited by:

    1. Xu, Yuanqing & Zheng, Xiaoxiao & Xin, Jie, 2022. "New non-traveling wave solutions for the (2+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    2. Lü, Xing & Chen, Si-Jia, 2023. "N-soliton solutions and associated integrability for a novel (2+1)-dimensional generalized KdV equation," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    3. 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.
    4. Yu, Weitian & Luan, Zitong & Zhang, Hongxin & Liu, Wenjun, 2022. "Collisions of three higher order dark double- and single-hump solitons in optical fiber," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    5. Kuo, Chun-Ku, 2021. "A study on the resonant multi-soliton waves and the soliton molecule of the (3+1)-dimensional Kudryashov–Sinelshchikov equation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    6. Sugati, Taghreed G. & Seadawy, Aly R. & Alharbey, R.A. & Albarakati, W., 2022. "Nonlinear physical complex hirota dynamical system: Construction of chirp free optical dromions and numerical wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    7. Rafiq, Muhammad Hamza & Raza, Nauman & Jhangeer, Adil, 2023. "Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    8. Yin, Yu-Hang & Lü, Xing, 2024. "Multi-parallelized PINNs for the inverse problem study of NLS typed equations in optical fiber communications: Discovery on diverse high-order terms and variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    9. 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.

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