Please wait a minute...
Chin. Phys. B, 2020, Vol. 29(10): 100501    DOI: 10.1088/1674-1056/ab9de0
General Prev   Next  

Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation

Xiangyu Yang(杨翔宇), Zhao Zhang(张钊), and Biao Li(李彪)†
1 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
Abstract  

Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent “phase shifts” of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.

Keywords:  soliton molecules      degenerate Darboux transformation      positons      phase shift      Gerdjikov-Ivanov equation  
Received:  13 February 2020      Revised:  05 June 2020      Accepted manuscript online:  18 June 2020
PACS:  05.45.Yv (Solitons)  
  02.30.Ik (Integrable systems)  
Corresponding Authors:  Corresponding author. E-mail: libiao@nbu.edu.cn   
About author: 
†Corresponding author. E-mail: libiao@nbu.edu.cn
* Project supported by the National Natural Science Foundation of China (Grant Nos. 11775121 and 11435005), and the K. C. Wong Magna Fund in Ningbo University.

Cite this article: 

Xiangyu Yang(杨翔宇), Zhao Zhang(张钊), and Biao Li(李彪)† Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation 2020 Chin. Phys. B 29 100501

Fig. 1.  

(a) Soliton molecule consisting of two solitons with parameter selections $ {\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6} $ , $ {\lambda }_{3}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4} $ , ξ = 40. (b) Soliton molecule consisting of three solitons with parameter selections $ {\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6} $ , $ {\lambda }_{3}=1+\displaystyle \frac{\sqrt{3}{\rm{i}}}{2} $ , $ {\lambda }_{5}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4} $ , ξ = 40. (c) Soliton molecule consisting of four solitons with parameter selections $ {\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6} $ , $ {\lambda }_{3}=1+\displaystyle \frac{\sqrt{3}{\rm{i}}}{2} $ , $ {\lambda }_{5}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4} $ , $ {\lambda }_{7}=\displaystyle \frac{4}{3}+\displaystyle \frac{\sqrt{55}{\rm{i}}}{6} $ , ξ = 10.

Fig. 2.  

The evolution of a two-positon |q2 − p| with α1 = 1 / 3, β1 = 2 / 5 of the GI equation: (a) 3D plot, (b) density plot, where two red curves are approximate trajectories defined by $ H\pm \displaystyle \frac{\mathrm{ln}(4096{\alpha }_{1}^{4}{\beta }_{1}^{4}{t}^{2})}{8{\alpha }_{1}{\beta }_{1}}=0 $ , which compared with density plot are shown consistence; (c) 2D plot of two-positon solution |q2 − p| at t = −100, t = 0, t = 100.

Fig. 3.  

The evolution of a three-positon |q3 − p| with α1 = 1 / 3, β1 = 2 / 5 of the GI equation: (a) 3D plot, (b) density plot, where two red curves are approximate trajectories defined by $ H\pm \displaystyle \frac{\mathrm{ln}(4194304{\alpha }_{1}^{8}{\beta }_{1}^{8}{t}^{4})}{8{\alpha }_{1}{\beta }_{1}} $ and the middle white curve is trajectory without phase shift, which compared with density plot are shown consistence; (c) 2D plot of three-positon solution |q3 − p| at t = −20, t = 0, t = 20.

Fig. 4.  

The evolution of a four-positon |q4 − p| with α1 = 1 / 2, β1 = 1 / 2 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.

Fig. 5.  

The evolution of hybrid solution consisting of a soliton and two-positon with α1 = 2 / 5, α1 = 1 / 5, α3 = 1 / 5, β3 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.

Fig. 6.  

The evolution of hybrid solution consisting of a soliton and three-positon with α1 = 2 / 5, β1 = 1 / 5, α3 = 1 / 5, β3 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.

Fig. 7.  

The evolution of hybrid solution consisting of two solitons and two-positon with α1 = 1 / 2, β1 = 1 / 2, α3 = 1 / 2, β3 = 1 / 3, α5 = 1 / 5, β5 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.

[1]
Johnson R S 1977 Proc. R. Soc. A 357 131 DOI: 10.1098/rspa.1977.0159
[2]
Anderson D, Lisak M 1983 Phys. Rev. A 27 1393 DOI: 10.1103/PhysRevA.27.1393
[3]
Clarkson P A, Tuszynski J A 1990 J. Phys. A: Math. Gen. 23 4269 DOI: 10.1088/0305-4470/23/19/013
[4]
Rogister A 1971 Phys. Fluids 14 2733 DOI: 10.1063/1.1693399
[5]
Kaup D J, Newell A C 1978 J. Math. Phys. 19 798 DOI: 10.1063/1.523737
[6]
Chen H H, Lee Y C, Liu C S 1979 Phys. Scr. 20 490 DOI: 10.1088/0031-8949/20/3-4/026
[7]
Gerdjikov V S, Ivanov I 1983 Bulg. J. Phys. 10 130
[8]
Kadkhoda N, Jafari H 2017 Optik 139 72 DOI: 10.1016/j.ijleo.2017.03.078
[9]
Ding C C, Gao Y T, Li L Q 2019 Chaos Solitons Fractals 120 259 DOI: 10.1016/j.chaos.2019.01.007
[10]
Fan E G 2000 J. Phys. A: Math. Gen. 33 6925 DOI: 10.1088/0305-4470/33/39/308
[11]
Xu S W, He J S 2012 J. Math. Phys. 53 063507 DOI: 10.1063/1.4726510
[12]
Guo L J, Zhang Y S, Xu S W, Wu Z W, He J S 2014 Phys. Scr. 89 035501 DOI: 10.1088/0031-8949/89/03/035501
[13]
Crasovan L, Kartashov Y, Mihalache D, Torner L, Kivshar Y, Pérez-García V 2003 Phys. Rev. E 67 046610 DOI: 10.1103/PhysRevE.67.046610
[14]
Weng W, Bouchand R, Lucas E, Obrzud E, Herr T, Kippenberg T J 2020 Nat. Commun. 11 2402 DOI: 10.1038/s41467-020-15720-z
[15]
Peng J S, Zeng H P 2018 Laser Photon. Rev. 12 1800009 DOI: 10.1002/lpor.201800009
[16]
Herink G, Kurtz F, Jalali B, Solli D R, Ropers C 2017 Science 356 50 DOI: 10.1126/science.aal5326
[17]
Liu X M, Yao X K, Cui Y D 2018 Phys. Rev. Lett. 121 023905 DOI: 10.1103/PhysRevLett.121.023905
[18]
Peng J S, Sorokina M, Sugavanam S, Tarasov N, Churkin D V, Turitsyn S K, Zeng H P 2018 Commun. Phys. 1 20 DOI: 10.1038/s42005-018-0022-7
[19]
Peng J S, Zeng H P 2019 Opt. Lett. 44 2899 DOI: 10.1364/OL.44.002899
[20]
Peng J S, Boscolo S, Zhao Z H, Zeng H P 2019 Sci. Adv. 5 eaax1110 DOI: 10.1126/sciadv.aax1110
[21]
Lou S Y 2020 J. Phys. Commun. 4 041002 DOI: 10.1088/2399-6528/ab833e
[22]
Zhang Z, Yang X Y, Li B 2020 Appl. Math. Lett. 103 106168 DOI: 10.1016/j.aml.2019.106168
[23]
Wang B, Zhang Z, Li B 2020 Chin. Phys. Lett. 37 030501 DOI: 10.1088/0256-307X/37/3/030501
[24]
Zhang Z, Yang S X, Li B 2019 Chin. Phys. Lett. 36 120501 DOI: 10.1088/0256-307X/36/12/120501
[25]
Matveev V B 1992 Phys. Lett. A 166 205 DOI: 10.1016/0375-9601(92)90362-P
[26]
Maisch H, Stahlhofen A A 1995 Phys. Scr. 52 228 DOI: 10.1088/0031-8949/52/3/002
[27]
Chow K W, Lai W C, Shek C K, Tso K 1998 Chaos Solitons Fractals 9 1901 DOI: 10.1016/S0960-0779(97)00128-8
[28]
Shao Y J, Zeng Y B 2005 J. Phys. A: Math. Gen. 38 2441 DOI: 10.1088/0305-4470/38/11/008
[29]
Dubard P, Gaillard P, Klein C, Matveev V B 2010 Eur. Phys. J. Spec. Top. 185 247 DOI: 10.1140/epjst/e2010-01252-9
[30]
Liu Y K, Li B 2017 Chin. Phys. Lett. 34 010202 DOI: 10.1088/0256-307X/34/1/010202
[31]
Qian C, Rao J G, Liu Y B, He J S 2016 Chin. Phys. Lett. 33 110201 DOI: 10.1088/0256-307X/33/11/110201
[32]
Silem A, Zhang C, Zhang D J 2019 Chin. Phys. B 28 020202 DOI: 10.1088/1674-1056/28/2/020202
[33]
Liu W, Zhang Y S, He J S 2018 Waves Random Complex Media 28 203 DOI: 10.1080/17455030.2017.1335916
[34]
Song W J, Xu S W, Li M H, He J S 2019 Nonliner Dyn. 97 2135 DOI: 10.1007/s11071-019-05111-5
[35]
Qiu D Q, Cheng W G 2019 Commun. Nonlinear Sci. Numer. Simulat. 78 104887 DOI: 10.1016/j.cnsns.2019.104887
[36]
Zhang Z, Yang X Y, Li W T, Li B 2019 Chin. Phys. B 28 110201 DOI: 10.1088/1674-1056/ab44a3
[37]
Peng J S, Zeng H P 2019 Commun. Phys. 2 34 DOI: 10.1038/s42005-019-0134-8
[38]
Lecaplain C, Grelu P, Soto-Crespo J M, Akhmediev N 2012 Phys. Rev. Lett. 108 233901 DOI: 10.1103/PhysRevLett.108.233901
[39]
Liu X M, Popa D, Akhmediev N 2019 Phys. Rev. Lett. 123 093901 DOI: 10.1103/PhysRevLett.123.093901
[40]
Zhang Z, Yang X Y, Li B 2020 Nonlinear Dyn. 100 1551 DOI: 10.1007/s11071-020-05570-1
[41]
Yang X Y, Fan R, Li B 2020 Phys. Scr. 95 045213 DOI: 10.1088/1402-4896/ab6483
[42]
Yan Z W, Lou S Y 2020 Appl. Math. Lett. 104 106271 DOI: 10.1016/j.aml.2020.106271
[43]
Liu X M, Pang M 2019 Laser Photon. Rev. 13 1800333 DOI: 10.1002/lpor.v13.9
[1] Soliton molecules, T-breather molecules and some interaction solutions in the (2+1)-dimensional generalized KDKK equation
Yiyuan Zhang(张艺源), Ziqi Liu(刘子琪), Jiaxin Qi(齐家馨), and Hongli An(安红利). Chin. Phys. B, 2023, 32(3): 030505.
[2] Three-step self-calibrating generalized phase-shifting interferometry
Yu Zhang(张宇). Chin. Phys. B, 2022, 31(3): 030601.
[3] Experimental demonstration of a fast calibration method for integrated photonic circuits with cascaded phase shifters
Junqin Cao(曹君勤), Zhixin Chen(陈志歆), Yaxin Wang(王亚新), Tianfeng Feng(冯田峰), Zhihao Li(李志浩), Zeyu Xing(邢泽宇), Huashan Li(李华山), and Xiaoqi Zhou(周晓祺). Chin. Phys. B, 2022, 31(11): 114204.
[4] Soliton molecules and asymmetric solitons of the extended Lax equation via velocity resonance
Hongcai Ma(马红彩), Yuxin Wang(王玉鑫), and Aiping Deng(邓爱平). Chin. Phys. B, 2022, 31(1): 010201.
[5] A low noise, high fidelity cross phase modulation in multi-level atomic medium
Liangwei Wang(王亮伟), Jia Guan(关佳), Chengjie Zhu(朱成杰), Runbing Li(李润兵), and Jing Shi(石兢). Chin. Phys. B, 2021, 30(11): 114204.
[6] Repulsive bubble-bubble interaction in ultrasonic field
Ling-Ling Zhang(张玲玲), Wei-Zhong Chen(陈伟中), Yao-Rong Wu(武耀蓉), Yang Shen(沈阳), and Guo-Ying Zhao(赵帼英). Chin. Phys. B, 2021, 30(10): 104301.
[7] A novel (2+1)-dimensional integrable KdV equation with peculiar solution structures
Sen-Yue Lou(楼森岳). Chin. Phys. B, 2020, 29(8): 080502.
[8] Generation of orbital angular momentum and focused beams with tri-layer medium metamaterial
Zhi-Chao Sun(孙志超), Meng-Yao Yan(闫梦瑶), and Bi-Jun Xu(徐弼军)†. Chin. Phys. B, 2020, 29(10): 104101.
[9] Single-shot phase-shifting digital holography with a photon-sieve-filtering telescope
You Li(李优), Yao-Cun Li(李垚村), Jun-Yong Zhang(张军勇), Yan-Li Zhang(张艳丽), Xue-Mei Li(李雪梅). Chin. Phys. B, 2019, 28(8): 084205.
[10] Simultaneous polarization separation and switching for 100-Gbps DP-QPSK signals in backbone networks
Yu-Long Su(苏玉龙), Huan Feng(冯欢), Hui Hu(胡辉), Wei Wang(汪伟), Tao Duan(段弢), Yi-Shan Wang(王屹山), Jin-Hai Si(司金海), Xiao-Ping Xie(谢小平), He-Ning Yang(杨合宁), Xin-Ning Huang(黄新宁). Chin. Phys. B, 2019, 28(2): 024216.
[11] Phase shift effects of radio-frequency bias on ion energy distribution in continuous wave and pulse modulated inductively coupled plasmas
Chan Xue(薛婵), Fei Gao(高飞), Yong-Xin Liu(刘永新), Jia Liu(刘佳), You-Nian Wang(王友年). Chin. Phys. B, 2018, 27(4): 045202.
[12] A new fully quantum-mechanical method used to calculate the collisional broadening coefficients and shift coefficients of Rb D1 lines perturbed by noble gases He and Ar
Wei Zhang(张伟), Yanchao Shi(史彦超), Bitao Hu(胡碧涛), Yi Zhang(张毅). Chin. Phys. B, 2018, 27(1): 013201.
[13] Performance analysis of quantum access network using code division multiple access model
Linxi Hu(胡林曦), Can Yang(杨灿), Guangqiang He(何广强). Chin. Phys. B, 2017, 26(6): 060304.
[14] Non-relativistic scattering amplitude for a new multi-parameter exponential-type potential
Yazarloo B H, Mehraban H, Hassanabadi H. Chin. Phys. B, 2016, 25(8): 080302.
[15] Self-calibration wavelength modulation spectroscopy for acetylene detection based on tunable diode laser absorption spectroscopy
Qin-Bin Huang(黄秦斌), Xue-Mei Xu(许雪梅), Chen-Jing Li(李晨静), Yi-Peng Ding(丁一鹏), Can Cao(曹粲), Lin-Zi Yin(尹林子), Jia-Feng Ding(丁家峰). Chin. Phys. B, 2016, 25(11): 114202.
No Suggested Reading articles found!