JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (5): 33-39.doi: 10.6040/j.issn.1671-9352.0.2024.175

Previous Articles    

Soliton solutions of the new(2+1)-dimensional Boussinesq equation

GUO Xuejun, CAO Yulei*   

  1. School of Mathematics and Physics, Nanyang Institute of Technology, Nanyang 473004, Henan, China
  • Published:2025-05-19

PDF (PC)

34

Abstract: The(2+1)-dimensional Boussinesq equation is investigated, which is a multidimensional extension of the typical Boussinesq equation. The soliton solutions of(2+1)-dimensional Boussinesq equation are constructed by using the Hirota bilinear method. The local characteristics of the soliton solutions are also analyzed, and the dynamic behaviors of these soliton solutions are given analytically. Additionally, under the limitation of special parameters, we also obtain the resonance soliton solutions, and due to the resonance collision, the soliton solution presents a "V" shape, which is no longer the traditional cross type. The dynamics of higher-order resonance soliton solutions are more complex and diverse, consisting of the superposition of fundamental resonant solitons.

Key words: (2+1)-dimensional Boussinesq equation, bilinear method, soliton solution, dynamics

CLC Number: 

  • O175.29
[1] HIROTA R. The direct method in soliton theory[M]. Cambridge: Cambridge University Press, 2004.
[2] 袁丰. 两个(2+1)维非线性发展方程的精确解[D]. 合肥:中国科学技术大学, 2020. YUAN Feng. Exact solutions of two(2+1)-dimensional nonlinear evolution equations[D]. Hefei: University of Science and Technology of China, 2020.
[3] 郭利娟. 两类Davey-Stewartson方程的有理解和半有理解[D]. 合肥:中国科学技术大学, 2019. GUO Lijuan. Rational and semi-rational solutions in two types of the Davey-Stewartson equation[D]. Hefei: University of Science and Technology of China, 2019.
[4] CAO Yulei, HE Jingsong, CHENG Yi. Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system[J]. Studies in Applied Mathematics, 2024, 152:648-672.
[5] CAO Yulei, CHENG Yi, HE Jingsong. Resonant collisions of high-order localized waves in the Maccari system[J]. Journal of Mathematical Physics, 2023, 64:043501.
[6] OHTA Y, YANG Jianke. General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation[J]. Proceedings of the Royal Society A, 2012, 468:1716-1740.
[7] RAO J J, FOKAS A S, HE J S. Doubly localized two-dimensional rogue waves in the davey-stewartson I equation[J]. Journal of Nonlinear Science, 2021, 31:67.
[8] HIROTA R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons[J]. Physical Review Letters, 1971, 27:1192.
[9] LI Yishen, GU Xinshen. Generating solution of Boussinesq equation by Darboux transformation of three order eigenvalue differential equations[J]. Ann Differential Equations, 1986, 4:419-422.
[10] GUO Lijuan, WANG Lihong, CHENG Yi, et al. Higher-order rogue waves and modulation instability of the two-component derivative nonlinear Schrödinger equation[J]. Communications in Nonlinear Science & Numerical Simulation, 2019, 79:104915.
[11] HE Jingsong, ZHANG H R, WANG Lihong, et al. Generating mechanism for higher-order rogue waves[J]. Physical Review E, 2013, 87:052914.
[12] GUO Boling, LING Liming, LIU Qingping. Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions[J]. Physical Review E, 2002, 85:026607.
[13] ZHANG Liangli, LU Xing, ZHU Shengzi. Painlevé Analysis, Bäcklund transformation and soliton solutions of the(2+1)-dimensional variable-coefficient Boussinesq Equation[J]. International Journal of Theoretical Physics, 2024, 63:160.
[14] LU Xing, ZHANG Liangli, MA Wenxiu. Oceanic shallow-water description with(2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito equation: Painlevé analysis, soliton solutions, and lump solutions[J]. Physics of Fluids, 2024, 36:064110.
[15] WANG Yi, LU Xing. Bäcklund transformation and interaction solutions of a generalized Kadomtsev-Petviashvili equation with variable coefficients[J]. Chinese Journal of Physics, 2024, 89:37-45.
[16] PENG Xue, ZHAO Yiwei, LU Xing. Data-driven solitons and parameter discovery to the(2+1)-dimensional NLSE in optical fiber communications[J]. Nonlinear Dynamics, 2024, 112:1291-1306.
[17] WAZWAZ A M, KAUR L. New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions[J]. Nonlinear Dynamics, 2019, 97:83-94.
[18] MILES J M. Resonantly interacting solitary waves[J]. Journal of Fluid Mechanics, 1977, 79:171-179.
[1] Wenyan LIU,Shuai QIAO,Chenghua GAO. Global dynamics analysis of a class of Filippov-type HR neuron model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(10): 13-23.
[2] BU Yuxiang, LUO Qi. Bistable solvated dielectron in liquid methylamine: intriguing spin crossover dynamics and dielectron exchanges [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(10): 113-126.
[3] Dong-cai CHENG,Zhi-xiong HUNAG,Run-tao REN,Jing-jing WANG,Zhang-ji YE,Liang ZHANG,Yue-ping WANG. Release of the antifoulant in equilibrium swollen polymer: molecular dynamics simulation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(11): 18-25, 34.
[4] CHEN Zhi-hui, WANG Zhen-zhen, CHENG Yong-kuan*. Soliton solutions for quasilinear Schrdinger equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(2): 58-62.
[5] PAN Long-qiang1,2, GENG Cun-liang1, MU Yu-guang3, LIU Xin4, HU Yi5, PAN Jing-shan4, ZHOU Ya-bin2, GONG Bin5, WANG Lu-shan1*. Deployment and optimization of biomacromolecule molecular dynamics simulation process on hundred trillion times cluster [J]. J4, 2012, 47(7): 14-19.
[6] WU Yong-gang1, SONG Han-feng1,2*, LI Yun1. Energy transfer in contact binaries [J]. J4, 2012, 47(1): 44-49.
[7] JU Lin. The two-soliton solution of a new shallow water wave equation [J]. J4, 2010, 45(11): 93-99.
[8] LI Xiao-jun,BAI Jin-tao, LI Yong-an, . The midpoint diagram of trajectory in billiard systems [J]. J4, 2008, 43(9): 36-41 .
[9] ZHANG Xing-gang,KONG Wei-shu .

Theoretical and numerical investigation of elastic string with fixed ends

[J]. J4, 2008, 43(10): 71-76 .
[10] LU Min,LUO Fei,LIU Wei-qing,WEI Wang-he . Molecular dynamics simulation on the structural stability of the Ag nanorod with ［111］ crystallographic orientation [J]. J4, 2008, 43(1): 43-48 .
[11] SHI Chang-guang . Multi-soliton solution of the Faddeev model [J]. J4, 2007, 42(7): 38-40 .
[12] XIE Jian-kun,YUE Qin-yan*,YU Hui,YUE Wen-wen,LI Ren-bo,ZHANG Sheng-xiao and WANG Xiao-na . Adsorptive properties of sludge activated carbon to Brilliant Red K-2BP [J]. J4, 2007, 42(3): 64-70 .
[13] ZHANG Xian-qiang,GUO W-ei-hua,YANG Ji-hong,LIU Cheng-cheng and WANG Ren-qing . A study of population structure and dynamics of Robinia pseudoacacia clones [J]. J4, 2006, 41(2): 135-139 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd