(2 + 1)维Hirota-Satsuma-Ito方程的怪波解及其动力学特征
Rogue Waves and Dynamics of Rogue Waves for the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation
摘要: 本文基于Kadomtsev-Petviashvili约化方法，推导出(2 + 1)维Hirota-Satsuma-Ito方程的一般高阶怪波解。这些高阶怪波解以Gram行列式形式呈现。通过对怪波解的动力学分析，发现该方程的解具有暗–亮波结构，并揭示了相关参数对波形的叠加与分离模式的调控机制。
Abstract: In this work, we derive general high-order rogue waves of the (2 + 1)-dimensional Hirota-Satsuma-Ito equation by Kadomtsev-Petviashvili hierarchy reduction technique. These general high order rogue waves are expressed in terms of Gram determinants. Through dynamical analysis of the rogue wave solutions, it is found that the rogue waves of this equation exhibit dark-bright wave structures. Moreover, the regulatory mechanism of relevant parameters on the superposition and separation patterns of the waveforms is revealed.
文章引用：王皎月, 吴文青. (2 + 1)维Hirota-Satsuma-Ito方程的怪波解及其动力学特征[J]. 理论数学, 2026, 16(4): 371-381. https://doi.org/10.12677/pm.2026.164122

参考文献

[1] Hopkin, M. (2004) Sea Snapshots Will Map Frequency of Freak Waves. Nature, 430, 492-492. [Google Scholar] [CrossRef] [PubMed]
[2] Kharif, C., Pelinovsky, E. and Slunyaev, A. (2008) Rogue Waves in the Ocean. Springer.
[3] Yeom, D. and Eggleton, B.J. (2007) Rogue Waves Surface in Light. Nature, 450, 953-954. [Google Scholar] [CrossRef] [PubMed]
[4] Solli, D.R., Ropers, C., Koonath, P. and Jalali, B. (2007) Optical Rogue Waves. Nature, 450, 1054-1057. [Google Scholar] [CrossRef] [PubMed]
[5] Bludov, Y.V., Konotop, V.V. and Akhmediev, N. (2009) Matter Rogue Waves. Physical Review A, 80, Article ID: 033610. [Google Scholar] [CrossRef] 
[6] Efimov, V.B., Ganshin, A.N., Kolmakov, G.V., McClintock, P.V.E. and Mezhov-Deglin, L.P. (2010) Rogue Waves in Superfluid Helium. The European Physical Journal Special Topics, 185, 181-193. [Google Scholar] [CrossRef] 
[7] Yan, Z. (2011) Vector Financial Rogue Waves. Physics Letters A, 375, 4274-4279. [Google Scholar] [CrossRef] 
[8] Peregrine, D.H. (1983) Water Waves, Nonlinear Schrödinger Equations and Their Solutions. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 25, 16-43. [Google Scholar] [CrossRef] 
[9] Shrira, V.I. and Geogjaev, V.V. (2010) What Makes the Peregrine Soliton So Special as a Prototype of Freak Waves? Journal of Engineering Mathematics, 67, 11-22. [Google Scholar] [CrossRef] 
[10] Chabchoub, A., Hoffmann, N., Onorato, M. and Akhmediev, N. (2012) Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves. Physical Review X, 2, Article ID: 011015. [Google Scholar] [CrossRef] 
[11] Tanaka, S. (1972) Modified Korteweg-Devries Equation and Scattering Theory. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 48, 466-469. [Google Scholar] [CrossRef] 
[12] Ablowitz, M.A. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press. [Google Scholar] [CrossRef] 
[13] Ablowitz, M.J. (2023) Nonlinear Waves and the Inverse Scattering Transform. Optik, 278, Article ID: 170710. [Google Scholar] [CrossRef] 
[14] Gu, C.H., Hu, H.S. and Zhou, Z.X. (2004) Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Springer.
[15] Guo, B., Ling, L. and Liu, Q.P. (2012) Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions. Physical Review E, 85, Article ID: 026607. [Google Scholar] [CrossRef] [PubMed]
[16] Mu, G., Qin, Z. and Grimshaw, R. (2015) Dynamics of Rogue Waves on a Multisoliton Background in a Vector Nonlinear Schrödinger Equation. SIAM Journal on Applied Mathematics, 75, 1-20. [Google Scholar] [CrossRef] 
[17] Hirota, R. (2004) The Direct Method in Soliton Theory. Cambridge University Press.
[18] Belokolos, E.D., Bobenko, A.I., Enolskii, V.Z., et al. (1994) Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer.
[19] Khater, A.H., El-Kalaawy, O.H. and Callebaut, D.K. (1998) Bäcklund Transformations and Exact Solutions for Alfvén Solitons in a Relativistic Electron-Positron Plasma. Physica Scripta, 58, 545-548. [Google Scholar] [CrossRef] 
[20] Ohta, Y. and Yang, J. (2012) General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schrödinger Equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 1716-1740. [Google Scholar] [CrossRef] 
[21] Chen, J., Chen, Y., Feng, B., Maruno, K. and Ohta, Y. (2018) General High-Order Rogue Waves of the (1 + 1)-Dimensional Yajima-Oikawa System. Journal of the Physical Society of Japan, 87, Article ID: 094007. [Google Scholar] [CrossRef] 
[22] Yang, B. and Yang, J. (2021) General Rogue Waves in the Three-Wave Resonant Interaction Systems. IMA Journal of Applied Mathematics, 86, 378-425. [Google Scholar] [CrossRef] 
[23] Feng, B., Shi, C., Zhang, G. and Wu, C. (2022) Higher-Order Rogue Wave Solutions of the Sasa-Satsuma Equation. Journal of Physics A: Mathematical and Theoretical, 55, Article ID: 235701. [Google Scholar] [CrossRef] 
[24] Zhang, G., Huang, P., Feng, B. and Wu, C. (2023) Rogue Waves and Their Patterns in the Vector Nonlinear Schrödinger Equation. Journal of Nonlinear Science, 33, Article No. 116. [Google Scholar] [CrossRef] 
[25] Wang, T., Qin, Z., Mu, G. and Zheng, F. (2023) General High-Order Rogue Waves in the Hirota Equation. Applied Mathematics Letters, 140, Article ID: 108571. [Google Scholar] [CrossRef] 
[26] Yang, B., Chen, J. and Yang, J. (2020) Rogue Waves in the Generalized Derivative Nonlinear Schrödinger Equations. Journal of Nonlinear Science, 30, 3027-3056. [Google Scholar] [CrossRef] 
[27] Mu, G., Zhang, C. and Yang, Z. (2025) Kadomtsev-petviashvili Reduction and Rational Solutions of the Generalized (2 + 1)-Dimensional Boussinesq Equation. Physics Letters A, 530, Article ID: 130125. [Google Scholar] [CrossRef] 
[28] Mu, G. and Qin, Z. (2014) Two Spatial Dimensional N-Rogue Waves and Their Dynamics in Mel’nikov Equation. Nonlinear Analysis: Real World Applications, 18, 1-13. [Google Scholar] [CrossRef] 
[29] Ye, L., Mu, G., Qin, Z., Yang, Z. and Feng, T. (2025) Rogue Waves and Lumps for a Generalized (3 + 1)-Dimensional Yu-Toda-Sasa-Fukuyama Equation in Fluids. Nonlinear Dynamics, 113, 27961-27979. [Google Scholar] [CrossRef] 
[30] Hirota, R. (1973) Exact n-Soliton Solutions of the Wave Equation of Long Waves in Shallow-Water and in Nonlinear Lattices. Journal of Mathematical Physics, 14, 810-814. [Google Scholar] [CrossRef] 
[31] Hirota, R. and Satsuma, J. (1976) n-Soliton Solutions of Model Equations for Shallow Water Waves. Journal of the Physical Society of Japan, 40, 611-612. [Google Scholar] [CrossRef] 
[32] Zhou, Y. and Manukure, S. (2019) Complexiton Solutions to the Hirota‐Satsuma‐Ito Equation. Mathematical Methods in the Applied Sciences, 42, 2344-2351. [Google Scholar] [CrossRef] 
[33] Liu, Y., Wen, X. and Wang, D. (2019) The n-Soliton Solution and Localized Wave Interaction Solutions of the (2 + 1)-Dimensional Generalized Hirota-Satsuma-Ito Equation. Computers & Mathematics with Applications, 77, 947-966. [Google Scholar] [CrossRef] 
[34] Kuo, C. and Ma, W. (2020) A Study on Resonant Multi-Soliton Solutions to the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equations via the Linear Superposition Principle. Nonlinear Analysis, 190, Article ID: 111592. [Google Scholar] [CrossRef] 
[35] Hong, X., Manafian, J., Ilhan, O.A., Alkireet, A.I.A. and Nasution, M.K.M. (2021) Multiple Soliton Solutions of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave. Journal of Geometry and Physics, 170, Article ID: 104338. [Google Scholar] [CrossRef] 
[36] Hossen, M.B., Towhiduzzaman, M., Harun-Or-Roshid, and Woadud, K.M.A.A. (2025) Mathematical Analysis of Shallow Water Wave and the Generalized Hirota-Satsuma-Ito Models: Soliton Solutions and Their Interactions. Results in Applied Mathematics, 28, Article ID: 100641. [Google Scholar] [CrossRef] 
[37] Liu, W., Wazwaz, A. and Zheng, X. (2019) High-order Breathers, Lumps, and Semi-Rational Solutions to the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation. Physica Scripta, 94, Article ID: 075203. [Google Scholar] [CrossRef] 
[38] Ma, W. (2019) Interaction Solutions to Hirota-Satsuma-Ito Equation in (2 + 1)-Dimensions. Frontiers of Mathematics in China, 14, 619-629. [Google Scholar] [CrossRef] 
[39] Yuan, F. and Ghanbari, B. (2024) A Study of Interaction Soliton Solutions for the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation. Nonlinear Dynamics, 112, 2883-2891. [Google Scholar] [CrossRef] 
[40] Liu, J., Zhu, W. and Zhou, L. (2020) Multi-Wave, Breather Wave, and Interaction Solutions of the Hirota-Satsuma-Ito Equation. The European Physical Journal Plus, 135, Article No. 20. [Google Scholar] [CrossRef] 
[41] Gong, Q.K., Wang, H. and Wang, Y.H. (2024) Localized Wave Solutions and Interactions of the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation. Chinese Physics B, 33, Article ID: 040505. [Google Scholar] [CrossRef] 
[42] Zhou, Y., Manukure, S. and Ma, W. (2019) Lump and Lump-Soliton Solutions to the Hirota-Satsuma-Ito Equation. Communications in Nonlinear Science and Numerical Simulation, 68, 56-62. [Google Scholar] [CrossRef] 
[43] Zhao, Z. and He, L. (2021) m-Lump and Hybrid Solutions of a Generalized (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation. Applied Mathematics Letters, 111, Article ID: 106612. [Google Scholar] [CrossRef] 
[44] Ma, W., Li, J. and Khalique, C.M. (2018) A Study on Lump Solutions to a Generalized Hirota‐Satsuma‐Ito Equation in (2 + 1)‐Dimensions. Complexity, 2018, Article ID: 9059858. [Google Scholar] [CrossRef] 
[45] Zhang, L., Tian, S., Peng, W., Zhang, T. and Yan, X. (2020) The Dynamics of Lump, Lumpoff and Rogue Wave Solutions of (2 + 1)-Dimensional Hirota-Satsuma-Ito Equations. East Asian Journal on Applied Mathematics, 10, 243-255. [Google Scholar] [CrossRef] 

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