|
|
|
Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation |
| Guofei Zhang(张国飞)1, Jingsong He(贺劲松)2,†, and Yi Cheng(程艺)1 |
1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
2 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China |
|
|
|
|
Abstract We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.
|
Received: 09 May 2022
Revised: 09 May 2022
Accepted manuscript online: 18 June 2022
|
|
PACS:
|
02.30.Ik
|
(Integrable systems)
|
| |
05.45.Yv
|
(Solitons)
|
|
| Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515012554). |
Corresponding Authors:
Jingsong He
E-mail: hejingsong@szu.edu.cn
|
Cite this article:
Guofei Zhang(张国飞), Jingsong He(贺劲松), and Yi Cheng(程艺) Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation 2022 Chin. Phys. B 31 110201
|
[1] Lax P D 1968 Commun. Pure Appl. Math. 21 467
[2] Ablowitz M J and Fokas A S 2003 Complex Variables: Introduction and Applications, (Cambridge: Cambridge University Press)
[3] Hasegawa A and Tappert F 1973 Appl. Phys. Lett. 23 142
[4] Hasegawa A and Kodama Y 1981 Proc. IEEE 69 1145
[5] Hasegawa A and Kodama Y 1995 Solitons in Optical Communication (Oxford: Oxford University Press)
[6] Turitsyn S K, Prilepsky J E, Le S T, Wahls S, Frumin L L, Kamalian M and Derevyanko S A 2017 Optica 4 307
[7] Biswas A 2005 Opt. Quantum Electron. 37 649
[8] Agalarov A, Zhulego V and Gadzhimuradov T 2015 Phys. Rev. E 91 042909
[9] Novikov S, Manakov S V, Pitaeskii L P and Zakharov V E 1984 Theory of soliton: the inverse scattering method (Berlin: Springer Science and Business Media)
[10] Coifman R R and Beals R 1984 Commun. Pure Appl. Math. 37 39
[11] Zhang G Q and Yan Z Y 2020 Physica D 410 132521
[12] Peng W Q and Chen Y 2022 J. Math. Phys. 63 033502
[13] Zhang G Q, Chen S Y and Yan Z Y 2020 Commun. Nonlinear Sci. Numer. Simul. 80 104927
[14] Pu J C and Chen Y 2021 arXiv:2105.06098 [nlin.SI]
[15] Zhu J Y and Chen Y 2022 J. Math. Phys. 62 123501
[16] Zhang G Q and Yan Z Y 2020 J. Nonlinear Sci. 30 3089
[17] Zhang X F, Tian S F and Yang J J 2021 Anal. Math. Phys. 11 1
[18] Wang X B and Han B 2020 J. Math. Anal. Appl. 487 123968
[19] Yang B and Chen Y 2019 Nonlinear Anal.-Real World Appl. 45 918
[20] Weng W F and Yan Z Y 2021 Mod. Phys. Lett. B 35 2150483
[21] Zhang Z Z anf Fan E G 2021 Z. Angew. Math. Phys. 72 153
[22] Mao J J, Tian S F, Xu T Z and Shi L F 2021 Nonlinear Dyn. 104 2639
[23] Zhang Y S, Tao X X, Yao T T and He J S 2020 Stud. Appl. Math. 145 812
[24] Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 190
[25] Johnson R S and Stewartson K 1977 Proc. R. Soc. Lond. A 357 131
[26] Subhadarshan S and Santanu S R 2017 J. Appl. Anal. Comput. 7 824
[27] Clarkson P A and Tuszynski J A 1990 J. Phys. A: Math. Gen. 23 4269
[28] Mjolhus E 1976 J. Plasma Phys. 16 321
[29] Chen H H, Lee Y C and Liu C S 1979 Phys. Scr. 20 490
[30] Gerdjikov V S and Ivanov I 1983 Bulg. J. Phys. 2 130
[31] Kaup D J and Newell A C 1978 J. Math. Phys. 19 798
[32] Hou Y, Fan E G and Zhao P 2013 J. Math. Phys. 54 073505
[33] Zhao P and Fan E G 2020 Physica D 402 132213
[34] Nie H, Zhu J Y and Geng X G 2018 Anal. Math. Phys. 8 415
[35] Takaoka H 2021 J. Differ. Equ. 291 90
[36] Wadati M, Konno K and Ichikawa Y 1979 J. Phys. Soc. Jpn. 46 1965
[37] Lin Y, Fang Y and Dong H 2019 Math. Probl. Eng. 2019 4058041
[38] Zhang B and Fan E G 2021 Mod. Phys. Lett. B 35 2150208
[39] Zhou X 1989 Commun. Pure Appl. Math. 42 895
[40] Biondini G and Kova?i? G 2014 J. Math. Phys. 55 031506 |
| No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
View more on Altmetrics
|
|
|
