Please wait a minute...
Chin. Phys. B, 2024, Vol. 33(2): 025203    DOI: 10.1088/1674-1056/ad082b
PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES Prev   Next  

Differences between two methods to derive a nonlinear Schrödinger equation and their application scopes

Yu-Xi Chen(陈羽西), Heng Zhang(张恒), and Wen-Shan Duan(段文山)
College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
Abstract  The present paper chooses a dusty plasma as an example to numerically and analytically study the differences between two different methods of obtaining nonlinear Schrödinger equation (NLSE). The first method is to derive a Korteweg-de Vries (KdV)-type equation and then derive the NLSE from the KdV-type equation, while the second one is to directly derive the NLSE from the original equation. It is found that the envelope waves from the two methods have different dispersion relations, different group velocities. The results indicate that two envelope wave solutions from two different methods are completely different. The results also show that the application scope of the envelope wave obtained from the second method is wider than that of the first one, though both methods are valuable in the range of their corresponding application scopes. It is suggested that, for other systems, both methods to derive NLSE may be correct, but their nonlinear wave solutions are different and their application scopes are also different.
Keywords:  dusty plasmas      nonlinear waves      particle-in-cell simulation  
Received:  30 August 2023      Revised:  29 October 2023      Accepted manuscript online:  31 October 2023
PACS:  52.27.Lw (Dusty or complex plasmas; plasma crystals)  
  52.65.Rr (Particle-in-cell method)  
  52.35.Mw (Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.))  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11965019 and 42004131) and the Foundation of Gansu Educational Committee (Grant No. 2022QB-178).
Corresponding Authors:  Heng Zhang, Wen-Shan Duan     E-mail:  zhangheng@nwnu.edu.cn;duanws@nwnu.edu.cn

Cite this article: 

Yu-Xi Chen(陈羽西), Heng Zhang(张恒), and Wen-Shan Duan(段文山) Differences between two methods to derive a nonlinear Schrödinger equation and their application scopes 2024 Chin. Phys. B 33 025203

[1] Ortoleva P and Ross J 1974 J. Chem. Phys. 60 5090
[2] Onorato M, Residori S, Bortolozzo U, Montina A and Arecchi F T 2013 Phys. Rep. 528 47
[3] Kharif C and Pelinovsky E 2003 Eur. J. Mech. B-Fluids 22 603
[4] Tien P K 1977 Rev. Mod. Phys. 49 361
[5] Figotin A and Vitebskiy I 2011 Laser Photon. Rev. 5 201
[6] Wang H Y and Wang X S 2016 J. Dyn. Differ. Equ. 28 143
[7] Anderson R R, Harvey C C, Hoppe M M, Tsurutani B T, Eastman T E and Etcheto J 1982 J. Geophys. Res.: Space Phys. 87 2087
[8] Gurnett D A and Frank L A 1978 J. Geophys. Res.: Space Phys. 83 1447
[9] Chau J L and Hysell D L 2004 Ann. Geophys. 22 4071
[10] Shukla P K, Kourakis I, Eliasson B, Marklund M and Stenflo L 2006 Phys. Rev. Lett. 97 094501
[11] Longuet-Higgins M S 1953 Philos. Trans. R. Soc. A 245 535
[12] Germain P, Masmoudi N and Shatah J 2012 Ann. Math. 175 691
[13] Chabchoub A, Hoffmann N, Onorato M and Akhmediev N 2012 Phys. Rev. X 2 011015
[14] Johnson R S 1980 J. Fluid Mech. 97 701
[15] Constantin A and Escher J 2007 Bull. Amer. Math. Soc. 44 423
[16] Constantin A 2010 Q. Appl. Math. 68 81
[17] Johnson R S 2002 J. Fluid Mech. 455 63
[18] Policastro G, Son D T and Starinets A O 2003 J. High Energy Phys. 2002 54
[19] Ruszkowski M, Brüggen M and Begelman M C 2004 Astrophys. J. 611 158
[20] Khatri R, Sunyaev R A and Chluba J 2012 Astron. Astrophys. 540 A124
[21] Buick J M, Buckley C L, Greated C A and Gilbert J 2000 J. Phys. A: Math. Gen. 33 3917
[22] Hassanien R H, Hou T Z, Li Y F and Li B M 2014 J. Integr. Agric. 13 335
[23] Lucas A 2016 Phys. Rev. B 93 245153
[24] Marin-Palomo P, Kemal J N, Karpov M, Kordts A, Pfeifle J, Pfeiffer M H, Trocha P, Wolf S, Brasch V, Anderson M H and others 2017 Nature 546 274
[25] Gauthier D J 2007 Nat. Photon. 1 92
[26] Honarasa G R, Hatami M and Tavassoly M K 2011 Commun. Theor. Phys. 56 322
[27] Uzunov I M, Stoev V D and Tzoleva T I 1993 Opt. Commun. 97 307
[28] Huang K 2018 Phys. Rev. E 97 052905
[29] Herrmann H J 1995 Chaos, Solitons Fractals 6 203
[30] Menotti C, Krämer M, Smerzi A, Pitaevskii L and Stringari S 2004 Phys. Rev. A 70 023609
[31] Fort C, Fallani L, Guarrera V, Lye J E, Modugno M, Wiersma D S and Inguscio M 2005 Phys. Rev. Lett. 95 170410
[32] Baizakov B B, Kamchatnov A M and Salerno M 2008 J. Phys. B: At., Mol. Opt. Phys. 41 215302
[33] Simula T P, Engels P, Coddington I, Schweikhard V, Cornell E A and Ballagh R J 2005 Phys. Rev. Lett. 94 080404
[34] Denschlag J, Simsarian J E, Feder D L, Clark C W, Collins L A, Cubizolles J, Deng L, Hagley E W, Helmerson K, Reinhardt W P and others 2000 Science 287 97
[35] Andrews M R, Kurn D M, Miesner H J, Durfee D S, Townsend C G, Inouye S and Ketterle W 1997 Phys. Rev. Lett. 79 553
[36] Nazarenko S and Onorato M 2006 Physica D 219 1
[37] Ankiewicz A, Bokaeeyan M and Akhmediev N 2019 Phys. Rev. E 99 050201
[38] Jeffrey A and Kakutani T 1972 Siam Rev. 14 582
[39] Osborne A R 1991 J. Comput. Phys. 94 284
[40] Israwi S and Kalisch H 2019 Phys. Lett. A 383 854
[41] Crabb M and Akhmediev N 2021 Phys. Rev. E 103 022216
[42] Johnson R S 1980 J. Fluid Mech. 97 701
[43] Gardner C S, Greene J M, Kruskal M D and Miura R M 1967 Phys. Rev. Lett. 19 1095
[44] Ankiewicz A, Wang Y, Wabnitz S and Akhmediev N 2014 Phys. Rev. E 89 012907
[45] Kedziora D J, Ankiewicz A and Akhmediev N 2012 Phys. Rev. E 85 066601
[46] Chabchoub A, Kibler B, Finot C, Millot G, Onorato M, Dudley J M and Babanin A V 2015 Ann. Phys. 361 490
[47] Mocz P and Succi S 2015 Phys. Rev. E 91 053304
[48] Li J and Li B 2021 Commun. Theor. Phys. 73 125001
[49] Edwards M and Burnett K 1995 Phys. Rev. A 51 1382
[50] Barletti L, Brugnano L, Caccia G F and Iavernaro F 2018 Appl. Math. Comput. 318 3
[51] Gill T S, Bains A S, Saini N S and Bedi C 2010 Phys. Lett. A 374 3210
[52] Lee N C 2012 Phys. Plasmas 19 082303
[53] Tang R A and Xue J K 2004 Phys. Plasmas 11 3939
[54] Goswami J and Sarkar J 2021 Phys. Scr. 96 085601
[55] Chaudhuri S and Chowdhury A R 2018 Chaos, Solitons Fractals 109 252
[56] Mishra M K, Chhabra R S and Sharma S R 1994 Phys. Plasmas 1 70
[57] Belmonte-Beitia J, Pérez-García V M, Vekslerchik V and Torres P J 2007 Phys. Rev. Lett. 98 064102
[58] Kivshar Y S and Malomed B A 1989 Rev. Mod. Phys. 61 763
[59] Shabat A and Zakharov V 1972 Sov. Phys. JETP 34 62
[60] Miura R M 1968 J. Math. Phys. 9 1202
[61] Gardner C S 1971 J. Math. Phys. 12 1548
[62] Schneider G 1998 J. Differ. Equations 147 333
[63] Özer M N and Taşcan F 2009 Chaos, Solitons Fractals 40 2265
[64] Fedele R and Schamel H 2002 Eur. Phys. J. B 27 313
[65] Yoshimura K and Watanabe S 1991 J. Phys. Soc. Jpn. 60 82
[66] Chabchoub A, Kibler B, Finot C, Millot G, Onorato M, Dudley J M and Babanin A V 2015 Ann. Phys. 361 490
[67] Changmai S and Bora M P 2020 Sci. Rep. 10 20980
[68] Tskhakaya D, Matyash K, Schneider R and Taccogna F 2007 Contrib. Plasma Phys. 47 563
[69] Gao D N, Zhang H, Zhang J, Li Z Z and Duan W S 2017 Phys. Plasmas 24 043703
[70] Qi X, Xu Y X, Duan W S and Yang L 2014 Phys. Plasmas 21 013702
[71] Zhang H, Qi X, Duan W S and Yang L 2015 Sci. Rep. 5 14239
[72] Zhang H, Yang Y, Hong X R, Qi X, Duan W S and Yang L 2017 Phys. Rev. E 95 053207
[73] Zhang J, Yang Y, Xu Y X, Yang L, Qi X and Duan W S 2014 Phys. Plasmas 21 103706
[74] Duan W S and Shi Y R 2003 Chaos, Solitons Fractals 18 321
[75] Wang F P, Han J f, Zhang J, Gao D N, Li Z Z, Duan W S and Zhang H 2018 Phys. Plasmas 25 032121
[76] Li Z Z and Duan W S 2021 Phys. Plasmas 28 043704
[77] Zhang H, Chen Y X, Wei L, Wang F P, Zhang W P and Duan W S 2023 J. Plasma Phys. 89 905890212
[78] Seadawy A R and Lu D 2017 Results Phys. 7 43
[79] Rahman M H, Chowdhury N A, Mannan A, Rahman M and Mamun A A 2018 Chin. J. Phys. 56 2061
[80] Sultana S and Kourakis I 2011 Plasma Phys. Controlled Fusion 53 045003
[81] Chowdhury N A, Mannan A, Hasan M M and Mamun A A 2019 Plasma Phys. Rep. 45 459
[1] Growth mechanism and characteristics of electron drift instability in Hall thruster with different propellant types
Long Chen(陈龙), Zi-Chen Kan(阚子晨), Wei-Fu Gao(高维富), Ping Duan(段萍), Jun-Yu Chen(陈俊宇), Cong-Qi Tan(檀聪琦), and Zuo-Jun Cui(崔作君). Chin. Phys. B, 2024, 33(1): 015203.
[2] Theoretical analyses on the one-dimensional charged particle transport in a decaying plasma under an electrostatic field
Yao-Ting Wang(汪耀庭), Xin-Li Sun(孙鑫礼), Lan-Yue Luo(罗岚月), Zi-Ming Zhang(张子明), He-Ping Li(李和平), Dong-Jun Jiang(姜东君), and Ming-Sheng Zhou(周明胜). Chin. Phys. B, 2023, 32(9): 095201.
[3] Intense low-noise terahertz generation by relativistic laser irradiating near-critical-density plasma
Shijie Zhang(张世杰), Weimin Zhou(周维民), Yan Yin(银燕), Debin Zou(邹德滨), Na Zhao(赵娜), Duan Xie(谢端), and Hongbin Zhuo(卓红斌). Chin. Phys. B, 2023, 32(3): 035201.
[4] Efficient ion acceleration driven by a Laguerre-Gaussian laser in near-critical-density plasma
Jia-Xiang Gao(高嘉祥), Meng Liu(刘梦), and Wei-Min Wang(王伟民). Chin. Phys. B, 2023, 32(10): 105202.
[5] Electron acceleration during magnetic islands coalescence and division process in a guide field reconnection
Shengxing Han(韩圣星), Huanyu Wang(王焕宇), and Xinliang Gao(高新亮). Chin. Phys. B, 2022, 31(2): 025202.
[6] Particle-in-cell simulation of ion-acoustic solitary waves in a bounded plasma
Lin Wei(位琳), Bo Liu(刘博), Fang-Ping Wang(王芳平), Heng Zhang(张恒), and Wen-Shan Duan(段文山). Chin. Phys. B, 2021, 30(3): 035201.
[7] Breather solutions of modified Benjamin-Bona-Mahony equation
G T Adamashvili. Chin. Phys. B, 2021, 30(2): 020503.
[8] Spontaneous growth of the reconnection electric field during magnetic reconnection with a guide field: A theoretical model and particle-in-cell simulations
Kai Huang(黄楷), Quan-Ming Lu(陆全明), Rong-Sheng Wang(王荣生), Shui Wang(王水). Chin. Phys. B, 2020, 29(7): 075202.
[9] Crystalline order and disorder in dusty plasmas investigated by nonequilibrium molecular dynamics simulations
Aamir Shahzad, Maogang He, Sheeba Ghani, Muhammad Kashif, Tariq Munir, Fang Yang. Chin. Phys. B, 2019, 28(5): 055201.
[10] Numerical simulation on modulational instability of ion-acoustic waves in plasma
Yi-Rong Ma(马艺荣), Lie-Juan Li(李烈娟), Wen-Shan Duan(段文山). Chin. Phys. B, 2019, 28(2): 025201.
[11] Small amplitude double layers in an electronegative dusty plasma with q-distributed electrons
Zhong-Zheng Li(李中正), Juan-Fang Han(韩娟芳), Dong-Ning Gao(郜东宁), Wen-Shan Duan(段文山). Chin. Phys. B, 2018, 27(10): 105204.
[12] Acceleration and radiation of externally injected electrons in laser plasma wakefield driven by a Laguerre-Gaussian pulse
Zhong-Chen Shen(沈众辰), Min Chen(陈民), Guo-Bo Zhang(张国博), Ji Luo(罗辑), Su-Ming Weng(翁苏明), Xiao-Hui Yuan(远晓辉), Feng Liu(刘峰), Zheng-Ming Sheng(盛政明). Chin. Phys. B, 2017, 26(11): 115204.
[13] Dynamic study of compressed electron layer driven by linearly polarized laser
Feng-chao Wang(王凤超). Chin. Phys. B, 2016, 25(5): 054102.
[14] Effect of inner-surface roughness of conical target on laser absorption and fast electron generation
Wang Huan (王欢), Cao Li-Hua (曹莉华), Zhao Zong-Qing (赵宗清), Yu Ming-Yang (郁明阳), Gu Yu-Qiu (谷渝秋), He Xian-Tu (贺贤土). Chin. Phys. B, 2014, 23(5): 055202.
[15] Effects of density profile and multi-species target on laser-heated thermal-pressure-driven shock wave acceleration
Wang Feng-Chao (王凤超). Chin. Phys. B, 2013, 22(12): 124102.
No Suggested Reading articles found!