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作为非线性发展方程的一种特殊局域解, 呼吸子具有包络振荡结构, 且这种振荡呈现周期性变化. 根据呼吸子在分布方向和演化方向的周期性, 呼吸子主要有3种类型, 即Kuznetsov-Ma呼吸子(Kuznetsov-Ma breather, KMB)、Akhmediev呼吸子(Akhmediev breather, AB)和一般呼吸子(general breather, GB). 近年来, 周期背景下的呼吸子现象在许多非线性物理领域被观察到, 比如在非线性光纤光学、流体力学等. 研究表明背景周期波的调制不稳定性可以激发呼吸子的产生, 且周期背景下的呼吸子具有非常丰富的物理性质和相互作用. 因此, 最近在周期背景下呼吸子的时空结构和相互作用引起了广泛关注. Gerdjikov-Ivanov方程可以被用来描述在量子场理论、弱非线性色散水波、非线性光学等领域中的非线性物理现象. 构造该模型的各种类型的解是非常有意义的工作. 据了解, 在椭圆函数背景下的多呼吸子之前还未被研究过. 本文首先利用修正的平方波(modified squared wave, MSW) 函数法和行波变换法获得该方程的椭圆函数解. 然后, 在椭圆函数解初始条件下得到该方程Lax对的通解. 基于椭圆函数的转换公式以及积分公式, 将势函数周期解化简为只含有Weierstrass椭圆函数. 然后, 利用达布变换构造出在椭圆函数背景下呼吸子的具体表达形式. 在椭圆函数背景下, 推导出3种不同类型的呼吸子, 包括GB, KMB和AB. 最后, 给出3种呼吸子的时空结构三维图, 并且展示它们之间相互作用的过程.
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关键词:
- Gerdjikov-Ivanov方程 /
- 椭圆函数 /
- 达布变换 /
- 呼吸子
As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.-
Keywords:
- Gerdjikov-Ivanov equation /
- elliptic function /
- Darboux transformation /
- breathers
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图 4 dn-周期波背景上的GB和KMB相互作用
Fig. 4. Interaction between GB and KMB on the dn-periodic wave background.
图 5 dn-周期波背景上的AB和KMB相互作用
Fig. 5. Interaction between AB and KMB on the dn-periodic wave background.
图 6 dn-周期波背景上的AB和GB相互作用
Fig. 6. Interaction between AB and GB on the dn-periodic wave background.
图 10 一般周期波背景上的两个GB相互作用
Fig. 10. Inacteraction between two GBs on the general periodic wave background.
图 11 一般周期波背景上的KMB和GB相互作用
Fig. 11. Inacteraction between KMB and GB on the general periodic wave background.
图 12 一般周期波背景上的AB和GB相互作用
Fig. 12. Interaction between AB and GB on the general periodic wave background.
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[1] Ablowitz M J, Satsuma J 1978 J. Math. Phys. 19 2180
Google Scholar
[2] Wazwaz A M 2009 Partial Differential Equations and Solitary Waves Theory (Berlin: Springer) pp285–413
[3] Zhou T Y, Tian B 2022 Appl. Math. Lett. 133 108280
Google Scholar
[4] Li B Q, Ma Y L 2020 Appl. Math. Comput. 386 125469
Google Scholar
[5] Kruglov V I, Triki H 2023 Chin. Phys. Lett. 40 090503
Google Scholar
[6] Hosseini K, Mirzazadeh M, Ilie M, Radmehr S 2020 Optik 206 164350
Google Scholar
[7] Vishnu Priya N, Senthilvelan M, Lakshmanan M 2013 Phys. Rev. E 88 022918
Google Scholar
[8] Kuznetsov E A 1977 Sov. Phys. Dokl. 22 507
[9] Ma Y C 1979 Stud. Appl. Math. 60 43
Google Scholar
[10] Akhmediev N, Komeev V I 1986 Theor. Math. Phys. 69 1089
Google Scholar
[11] Its A R, Rybin A V, Sall M A 1988 Theor. Math. Phys. 74 20
Google Scholar
[12] Walczak P, Randoux S, Suret P 2015 Phys. Rev. Lett. 114 143903
Google Scholar
[13] Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502
Google Scholar
[14] Xiong H, Gan J H, Wu Y 2017 Phys. Rev. Lett. 119 153901
Google Scholar
[15] Ding C C, Zhou Q, Xu L, Triki H, Mirzazadeh M, Liu W J 2023 Chin. Phys. Lett. 40 040501
Google Scholar
[16] 黎旭君 2018 硕士学位论文(武汉: 武汉大学)
Li X J 2018 M. S. Thesis (Wuhan: Wuhan University
[17] Kaup D J, Newell A C 1978 J. Math. Phys. 19 798
Google Scholar
[18] Chen H H, Lee Y C, Liu C S 1979 Phys. Scr. 20 490
Google Scholar
[19] Liu C, Wu Y H, Chen S C, Yao X K, Akhmediev N 2021 Phys. Rev. Lett. 127 094102
Google Scholar
[20] Gerdzhikov V S, Ivanov M I 1982 Theor. Math. Phys. 52 676
[21] Ji T, Zhai Y Y 2020 Nonlinear Dyn. 101 619
Google Scholar
[22] Zhang Z C, Fan E G 2021 Z. Angew. Math. Phys. 72 153
Google Scholar
[23] Lou Y, Zhang Y, Ye R S, Li M 2021 Wave Motion 106 102795
Google Scholar
[24] Fan E G 2000 J. Math. Phys. 41 7769
Google Scholar
[25] Dai H H, Fan E G 2004 Chaos Solitons Fractals 22 93
Google Scholar
[26] Xu S W, He J S 2012 J. Math. Phys. 53 063507
[27] Guo L J, Zhang Y S, Xu S W, Wu Z W, He J S 2014 Phys. Scr. 89 035501
Google Scholar
[28] Kedziora D J, Ankiewicz A, Akhmediev N 2014 Eur. Phys. J. Spec. Top. 223 43
Google Scholar
[29] Chen J B, Pelinovsky D E 2018 Proc. R. Sic. A 474 20170814
Google Scholar
[30] Leykam D, Smolina E, Maluckov A, Flach S, Smirnova D A 2021 Phys. Rev. Lett. 126 073901
Google Scholar
[31] Chen S C, Liu C, Akhmediev N 2023 Phys. Rev. A 107 063507
Google Scholar
[32] Liu C, Chen S C, Yao X K, Akhmediev N 2022 Chin. Phys. Lett. 39 094201
Google Scholar
[33] Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526
Google Scholar
[34] Chen S C, Liu C 2022 Physica D 438 133364
Google Scholar
[35] Che W J, Liu C, Akhmediev N 2023 Phys. Rev. E 107 054206
Google Scholar
[36] Liu C, Chen S C, Yao X K, Akhmediev N 2022 Physica D 433 133192
Google Scholar
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