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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.
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Keywords:
- Gerdjikov-Ivanov equation /
- elliptic function /
- Darboux transformation /
- breathers
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图 1 dn-周期波背景上的AB
Figure 1. AB on the dn-periodic wave background.
图 2 dn-周期波背景上的KMB
Figure 2. KMB on the dn-periodic wave background.
图 3 dn-周期波背景上的GB
Figure 3. GB on the dn-periodic wave background.
图 4 dn-周期波背景上的GB和KMB相互作用
Figure 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
Figure 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
Figure 6. Interaction between AB and GB on the dn-periodic wave background.
图 7 一般周期波背景上的AB
Figure 7. AB on the general periodic wave background.
图 8 一般周期波背景上的KMB
Figure 8. KMB on the general periodic wave background.
图 9 一般周期波背景上的GB
Figure 9. GB on the general periodic wavebackground.
图 10 一般周期波背景上的两个GB相互作用
Figure 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
Figure 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
Figure 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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