Hirota 方程的光孤子共振和孤子分子

摘要/Abstract

摘要：

孤子共振和孤子分子对光学、物理和流体力学等领域的研究具有重要意义. 该文在非线性光学系统的背景下, 对 Hirota 方程的光孤子共振和孤子分子特性进行了系统性研究. 首先, 基于 Lax 对构造出 Hirota 方程的 $N$ 重 Darboux 变换. 并由此推导出 $N$ 孤子解的精确解析表达式. 接下来, 通过改变谱参数和初始相位参数, 构造出 Hirota 方程的新型孤子共振和孤子分子解, 并分析其动力学特性. 研究结果表明, 局域孤子共振可演化为孤子分子, 孤子分子可视为局域孤子共振的极限形式. 此外, 研究揭示了孤子携带的能量在孤子共振与孤子分子形成过程中呈现增长趋势, 并表明孤子分子的时空演化模式随孤子阶数的提升而趋于复杂. 同时, 孤子分子的波密度与高阶色散项及高阶非线性项的系数呈正相关关系. 上述结果为 Hirota 方程孤子动力学的深入研究提供了新的理论依据, 并有助于理解非线性光学系统中的孤子相互作用机制.

关键词: Hirota 方程, Darboux 变换, 孤子共振, 孤子分子

Abstract:

Soliton resonance and soliton molecules play a significant role in the study of optics, physics, and fluid mechanics. This paper systematically investigates the soliton resonance and soliton molecule properties of the Hirota equation in the context of nonlinear optical systems. First, based on the Lax pair, the $N$-fold Darboux transformation of the Hirota equation is constructed, from which the exact analytical expressions of the $N$-soliton solutions are derived. Subsequently, by adjusting the spectral parameters and initial phase parameters, novel soliton resonance and soliton molecule solutions of the Hirota equation are constructed, and their dynamical characteristics are analyzed. The research results indicate that localized soliton resonance can evolve into soliton molecules, with soliton molecules being the limiting form of localized soliton resonance. Furthermore, the study reveals that the energy carried by solitons exhibits a growth trend during the formation of soliton resonance and soliton molecules. It also demonstrates that the spatiotemporal evolution patterns of soliton molecules become increasingly complex as the soliton order increases. Meanwhile, the wave density of soliton molecules is positively correlated with the coefficients of higher-order dispersion and higher-order nonlinear terms. These findings provide new theoretical insights into the soliton dynamics of the Hirota equation and contribute to a better understanding of soliton interaction mechanisms in nonlinear optical systems.

Key words: Hirota equation, Darboux transformation, soliton resonances, soliton molecules

中图分类号:

O175.29

曾平平, 曾平安, 刘露. Hirota 方程的光孤子共振和孤子分子[J]. 数学物理学报, 2026, 46(1): 238-248.

Pingping Zeng, Pingan Zeng, Lu Liu. Optical Soliton Resonances and Soliton Molecules for the Hiorta Equation[J]. Acta mathematica scientia,Series A, 2026, 46(1): 238-248.

图/表 8

图1

图2

图3

图4

图5

图6

图7

图8

参考文献 38

[1]	Kadomtsev B B, Petviashvili V I. On the stability of solitary waves in weakly dispersing media. Sov Phys Dokl, 1970, 192: 753-756
[2]	Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett, 1971, 27(18): Art 1192
[3]	Cox S M, Matthews P C. Exponential time differencing for stiff systems. J Comput Phys, 2002, 176(2): 430-455 doi: 10.1006/jcph.2002.6995
[4]	Bamieh B, Paganini F, Dahleh M A. Distributed control of spatially invariant systems. IEEE Trans Autom Control, 2002, 47(7): 1091-1107 doi: 10.1109/TAC.2002.800646
[5]	Shu C W. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev, 2009, 51(1): 82-126 doi: 10.1137/070679065
[6]	郭艳凤, 崔静易, 肖海军, 张景军. 新 (3 + 1) 维 KP 方程的退化解与相互作用解. 数学物理学报, 2024, 44A(6): 1520-1536
	Guo Y F, Cui J Y, Xiao H J, Zhang J J. Degeneration behaviors of solutions and hybrid solutions for the new (3+1)-dimensional KP equation. Acta Math Sci, 2024, 44A(6): 1520-1536
[7]	Nkenfack C E, Lekeufack O T, Yamapi R, Kengne E. Bright solitons and interaction in the higher-order Gross-Pitaevskii equation investigated with Hirota's bilinear method. Phys Lett A, 2024, 511: Art 129563
[8]	Liu T S, Xia T C. Multi-component generalized Gerdjikov-Ivanov integrable hierarchy and its Riemann-Hilbert problem. Nonlinear Anal Real World Appl, 2022, 68: Art 103667
[9]	Ma W X. Riemann-Hilbert problems and soliton solutions of nonlocal reverse-time NLS hierarchies. Acta Math Sci, 2022, 42B(1): 127-140
[10]	Ma L Y, Li B Q. Dynamics of soliton resonances and soliton molecules for the AB system in two-layer fluids. Nonlinear Dynam, 2023, 111(14): 13327-13341 doi: 10.1007/s11071-023-08529-0
[11]	Wang K L, Huang L, Yu J. Darboux transformation and soliton solutions of the coupled generalized Sasa-Satsuma equation. Nonlinear Dynam, 2023, 111(22): 21279-21288 doi: 10.1007/s11071-023-08944-3
[12]	Feng L L, Zhu Z N. Darboux transformation and soliton solutions for a nonlocal two-component complex modified Korteweg-de Vries equation. J Nonlinear Math Phys, 2023, 30(3): 1011-1031 doi: 10.1007/s44198-023-00112-w
[13]	Wu X H, Gao Y T, Yu X, Liu F Y. Generalized Darboux transformation and solitons for a Kraenkel-Manna-Merle system in a ferromagnetic saturator. Nonlinear Dynam, 2023, 111(15): 14421-14433 doi: 10.1007/s11071-023-08510-x
[14]	Chen J B, Pelinovsky D E. Bright and dark breathers of the Benjamin-Ono equation on the traveling periodic background. Wave Motion, 2024, 126: Art 103263
[15]	Liu S H, Tian B, Gao X T. Lax pair, conservation laws, breather-to-soliton transitions and modulation instability for a coupled extended modified Korteweg-de Vries system in a fluid. Eur Phys J Plus, 2024, 139(6): Art 496
[16]	王秀彬, 田守富. 耦合 Sasa-Satsuma 方程的 Darboux-穿衣变换和半有理解. 应用数学学报, 2024, 47(1): 124-138 doi: 10.20142/j.cnki.amas.202401009
	Wang X B, Tian S F. Darboux-dressing transformation and semirational solution to the coupled Sasa-Satsuma equation. Acta Math Appl Sin, 2024, 47(1): 124-138 doi: 10.20142/j.cnki.amas.202401009
[17]	Wang Z Y, Tian S F, Yang J J. Riemann-Hilbert problem and multiple high-order poles solutions of the focusing mKdV equation with nonzero boundary conditions. Acta Math Appl Sin Engl Ser, 2025, 41(1): 234-251 doi: 10.1007/s10255-024-1037-3
[18]	Rotschild C, Alfassi B, Cohen O, Segev M. Long-range interactions between optical solitons. Nat Phys, 2006, 2(11): 769-774 doi: 10.1038/nphys445
[19]	Oktem B, Ulgudur C, Ilday F O. Soliton-similariton fibre laser. Nat Photonics, 2010, 4(5): 307-311 doi: 10.1038/nphoton.2010.33
[20]	Zhao L C, Ling L M, Yang Z T, Liu J. Properties of the temporal-spatial interference pattern during soliton interaction. Nonlinear Dynam, 2016, 83: 659-665 doi: 10.1007/s11071-015-2354-0
[21]	Kuprikov E, Kokhanovskiy A, Serebrennikov K, Turitsyn S. Deep reinforcement learning for self-tuning laser source of dissipative solitons. Sci Rep, 2022, 12(1): Art 7185
[22]	Han Y, Gao B, Wen H L, et al. Pure-high-even-order dispersion bound solitons complexes in ultra-fast fiber lasers. Light Sci Appl, 2024, 13: Art 101
[23]	Li C Y, Kartashov Y V. Stable vortex solitons sustained by localized gain in a cubic medium. Phys Rev Lett, 2024, 132(21): Art 213802
[24]	Zhou Q. Influence of parameters of optical fibers on optical soliton interactions. Chin Phys Lett, 2022, 39(1): Art 010501
[25]	Zhou Q, Wang T Y, Biswas A, Liu W J. Nonlinear control of logic structure of all-optical logic devices using soliton interactions. Nonlinear Dynam, 2022, 107: 1215-1222 doi: 10.1007/s11071-021-07027-5
[26]	Gao Y T, Tian B. On the non-planar dust-ion-acoustic waves in cosmic dusty pal smas with transverse perturbations. EPL, 2007, 77(1): Art 15001
[27]	张岩, 郝惠琴, 郭睿. 阶跃初值条件解的完全分类: 流体力学中广义 Gardner 方程的分析与数值验证. 数学物理学报, 2024, 44A(5): 1242-1282
	Zhang Y, Hao H Q, Guo R. The complete classification of solutions to the step initial condition: Analysis and numerical verification for the generalized Gardner equation in fluid mechanics. Acta Math Sci, 2024, 44A(5): 1242-1282
[28]	Chang W, Ankiewicz A, Soto-Crespo J M, Akhmediev N. Dissipative soliton resonances. Phys Rev A, 2008, 78(2): Art 023930
[29]	Wu X, Tang D Y, Zhang H, Zhao L M. Dissipative soliton resonance in an all-normal-dispersion erbium-doped fiber laser. Opt Exp, 2009, 17(7): 5580-5584 doi: 10.1364/OE.17.005580
[30]	Grelu P, Belhache F, Gutty F, Soto-Crespo J M. Phase-locked soliton pairs in a strtched-pulse fiber laser. Opt Lett, 2002, 27(11): 966-968 pmid: 18026339
[31]	Jia M, Lin J, Lou S Y. Soliton and breather molecules in few-cycle-pulse optical model. Nonlinear Dynam, 2020, 100: 3745-3757 doi: 10.1007/s11071-020-05695-3
[32]	Wang B, Zhang Z, Li B A. Soliton molecules and some hybrid solutions for the nonlinear Schr$\rm\ddot{o}$dinger equation. Chin Phys Lett. 2020, 37(3): Art 030501
[33]	Li B Q, Ma Y L. Soliton resonances and soliton molecules of pump wave and Stokes wave for a transient stimulated Raman scattering system in optics. Eur Phys J Plus, 2022, 137(11): Art 1227
[34]	Li B Q, Ma Y L. Soliton resonances and soliton molecules for the Lakshmanan-Porsezian-Daniel system in optical fibers. Nonlinear Dynam, 2023, 111(7): 6689-6699 doi: 10.1007/s11071-022-08195-8
[35]	Hirota R. Exact envelope-soliton solutions of a nonlinear wave equation. J Math Phys, 1973, 14(7): 805-809 doi: 10.1063/1.1666399
[36]	Ankiewicz A, Soto-Crespo J M, Akhmediev N. Rogue waves and rational solutions of the Hirota equation. Phys Rev E, 2010, 81(4): Art 046602
[37]	Tao Y S, He J S. Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys Rev E, 2012, 85(2): Art 026601
[38]	Liu D Y, Yu H M. Mixed localized wave solutons of the Hirota equation. Appl Math Lett, 2021, 118: Art 107154