Korteweg-de Vries方程的准孤立子解及其在离子声波中的应用

王建勇; 程雪苹; 曾莹; 张元祥; 葛宁怡

doi:10.7498/aps.67.20180094

摘要

应用推广的tanh函数展开法，给出了Korteweg-de Vries方程具有准孤立子行为的两组孤子-椭圆周期波解，其中一组为新解.推导了均匀磁化等离子体中描述离子声波动力学行为的Korteweg-de Vries方程，发现电子分布、离子电子温度比、磁场大小、磁场方向对离子声准孤立子的波形具有显著影响.

关键词:

Investigation of interaction between solitons and their background small amplitude waves has been an interesting topic in numerical study for more than three decades. A classical soliton accompanied with oscillatory tails to infinite extent in space, is an interesting quasi-soliton, which has been revealed in experimental study and really observed. However, analytical solution of such a special quasi-soliton structure is rarely considered. In this paper, two branches of soliton-cnoidal wave solution as well as the two-soliton solution of the Korteweg-de Vries (KdV) equation are obtained by the generalized tanh expansion method. The exact relation between the soliton-cnoidal wave solution and the classical soliton solution of the KdV equation is established. By choosing suitable wave parameters, the quasi-soliton behavior of the soliton-cnoidal wave solution is revealed. It is found that with modulus of the Jacobi elliptic function approaching to zero asymptotically, the oscillating tails can be minimized and the soliton core of the soliton-cnoidal wave turns closer to the classical soliton solution. In addition, the quasi-soliton structure is revealed in a plasma physics system. By the reductive perturbation approach, the KdV equation modeling ion acoustic waves in an ideal homogeneous magnetized plasma is derived. It is confirmed that the waveform of the quasi-soliton is significantly influenced by the electron distribution, temperature ratio of ion to electron, magnetic field strength, and magnetic field direction. Interestingly, the amplitude of the quasi-soliton keeps constant due to the -independence of nonlinear coefficient A. The width of the soliton core and the wavelength of the surrounded periodic wave become constant with the further increase of . The explicit soliton-cnoidal wave solution with quasi-soliton behavior obtained here is applicable to many physical scenarios. For instance, the quasi-soliton structure can be viewed as a classical soliton with perturbations, and can correct the classical soliton in both theoretical and experimental study.

Keywords:

作者及机构信息

1.
衢州学院教师教育学院, 衢州 324000;

2.
浙江海洋大学物理系, 舟山 316004;

3.
衢州学院机械工程学院, 衢州 324000

通信作者: 王建勇, jywangqz@126.com

基金项目: 国家自然科学基金（批准号：11605102，11505154，51605252）和衢州学院博士科研启动基金（批准号：201507，201508）资助的课题.

Authors and contacts

1.
College of Teacher Education, Quzhou University, Quzhou 324000, China;

2.
Department of Physics, Zhejiang Ocean University, Zhoushan 316004, China;

3.
College of Mechanical Engineering, Quzhou University, Quzhou 324000, China

Corresponding author: Wang Jian-Yong, jywangqz@126.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11605102, 11505154, 51605252) and the Doctoral Starting up Foundation of Quzhou University, China (Grant Nos. 201507, 201508).

参考文献

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[2]	Miura R, Gardner C, Kruskal M 1968 J. Math. Phys. 9 1204
[3]	Hirota R 1971 Phys. Rev. Lett. 27 1192
[4]	Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 25 522
[5]	Dauxois T, Peyrard M 2006 Physics of Solitons (Cambridge: Cambridge University Press)
[6]	Jeffrey A, Kakutani T 1972 SIAM Rev. 14 582
[7]	Bandyopadhyay P, Prasad G, Sen A, Kaw P K 2008 Phys. Rev. Lett. 101 065006
[8]	Cheng X P, Li J Y, Xue J R 2011 Acta Phys. Sin. 60 110204(in Chinese) [程雪苹, 李金玉, 薛江蓉 2011 物理学报 60 110204]
[9]	Mao J J, Yang J R, Li C Y 2012 Acta Phys. Sin. 61 020206(in Chinese) [毛杰健, 杨建荣, 李超英 2012 物理学报 61 020206]
[10]	Lou S Y, Hu X R, Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[11]	Lou S Y 2015 Stud. Appl. Math. 134 372
[12]	Tang X Y, Hao X Z, Liang Z F 2017 Comp. Math. Appl. 74 1311
[13]	Tang X Y, Liang Z F, Wang J Y 2015 J. Phys. A: Math. Theor. 48 285204
[14]	Gao X N, Lou S Y, Tang X Y 2013 JHEP 05 029
[15]	Chen C L, Lou S Y 2013 Chin. Phys. Lett. 30 110202
[16]	Cheng X P, Lou S Y, Chen C L, Tang X Y 2014 Phys. Rev. E 89 043202
[17]	Ren B, Cheng X P, Lin J 2016 Nonlinear Dyn. 86 1855
[18]	Ren B 2017 Commun. Nonlinear Sci. Numer. Simul. 42 456
[19]	Hao X Z, Liu Y P, Tang X Y, Li Z B 2016 Comp. Math. Appl. 72 2405
[20]	Wang J Y, Cheng X P, Tang X Y, Yang J R, Ren B 2014 Phys. Plasmas 21 032111
[21]	Deeskow P, Schamel H, Rao N N, Yu M Y, Varma R K, Shukla P K 1987 Phys. Fluids 30 2703
[22]	Keane A J, Mushtaq A, Wheatland M S 2011 Phys. Rev. E 83 066407
[23]	Davis R E, Acrivos A 1967 J. Fluid Mech. 29 593
[24]	Farmer D M, Smith J D 1980 Deep-sea Rea. 27A 239
[25]	Akylas T R, Grimshaw R H J 1992 J. Fluid Mech. 242 279
[26]	Wang J Y, Tang X Y, Lou S Y, Gao X N, Jia M 2014 Europhys. Lett. 108 20005
[27]	Williams G, Kourakis I 2013 Plasma Phys. Controlled Fusion 55 055005
[28]	Singh S V, Devanandhan S, Lakhina G S, Bharuthram R 2013 Phys. Plasmas 20 012306
[29]	Saini N S, Kourakis I 2010 Plasma Phys. Controlled Fusion 52 075009

施引文献

[1]	Lax P D 1968 Commun. Pur. Appl. Math. 21 467
[2]	Miura R, Gardner C, Kruskal M 1968 J. Math. Phys. 9 1204
[3]	Hirota R 1971 Phys. Rev. Lett. 27 1192
[4]	Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 25 522
[5]	Dauxois T, Peyrard M 2006 Physics of Solitons (Cambridge: Cambridge University Press)
[6]	Jeffrey A, Kakutani T 1972 SIAM Rev. 14 582
[7]	Bandyopadhyay P, Prasad G, Sen A, Kaw P K 2008 Phys. Rev. Lett. 101 065006
[8]	Cheng X P, Li J Y, Xue J R 2011 Acta Phys. Sin. 60 110204(in Chinese) [程雪苹, 李金玉, 薛江蓉 2011 物理学报 60 110204]
[9]	Mao J J, Yang J R, Li C Y 2012 Acta Phys. Sin. 61 020206(in Chinese) [毛杰健, 杨建荣, 李超英 2012 物理学报 61 020206]
[10]	Lou S Y, Hu X R, Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[11]	Lou S Y 2015 Stud. Appl. Math. 134 372
[12]	Tang X Y, Hao X Z, Liang Z F 2017 Comp. Math. Appl. 74 1311
[13]	Tang X Y, Liang Z F, Wang J Y 2015 J. Phys. A: Math. Theor. 48 285204
[14]	Gao X N, Lou S Y, Tang X Y 2013 JHEP 05 029
[15]	Chen C L, Lou S Y 2013 Chin. Phys. Lett. 30 110202
[16]	Cheng X P, Lou S Y, Chen C L, Tang X Y 2014 Phys. Rev. E 89 043202
[17]	Ren B, Cheng X P, Lin J 2016 Nonlinear Dyn. 86 1855
[18]	Ren B 2017 Commun. Nonlinear Sci. Numer. Simul. 42 456
[19]	Hao X Z, Liu Y P, Tang X Y, Li Z B 2016 Comp. Math. Appl. 72 2405
[20]	Wang J Y, Cheng X P, Tang X Y, Yang J R, Ren B 2014 Phys. Plasmas 21 032111
[21]	Deeskow P, Schamel H, Rao N N, Yu M Y, Varma R K, Shukla P K 1987 Phys. Fluids 30 2703
[22]	Keane A J, Mushtaq A, Wheatland M S 2011 Phys. Rev. E 83 066407
[23]	Davis R E, Acrivos A 1967 J. Fluid Mech. 29 593
[24]	Farmer D M, Smith J D 1980 Deep-sea Rea. 27A 239
[25]	Akylas T R, Grimshaw R H J 1992 J. Fluid Mech. 242 279
[26]	Wang J Y, Tang X Y, Lou S Y, Gao X N, Jia M 2014 Europhys. Lett. 108 20005
[27]	Williams G, Kourakis I 2013 Plasma Phys. Controlled Fusion 55 055005
[28]	Singh S V, Devanandhan S, Lakhina G S, Bharuthram R 2013 Phys. Plasmas 20 012306
[29]	Saini N S, Kourakis I 2010 Plasma Phys. Controlled Fusion 52 075009

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