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

[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 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 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 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 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

[1]	岳东宁, 董全力, 陈民, 赵耀, 耿盼飞, 远晓辉, 盛政明, 张杰. 强激光与近临界密度等离子体相互作用中的无碰撞静电冲击波产生. , 2023, 72(11): 115202. doi: 10.7498/aps.72.20230271
[2]	李沁然, 孙超, 谢磊. 浅海内孤立波动态传播过程中声波模态强度起伏规律. , 2022, 71(2): 024302. doi: 10.7498/aps.71.20211132
[3]	李沁然, 孙超, 谢磊. 浅海内孤立波动态传播过程中声波模态强度起伏规律研究. , 2021, (): . doi: 10.7498/aps.70.20211132
[4]	胡广海, 金晓丽, 张乔枫, 谢锦林, 刘万东. 利用离子声波朗道阻尼测量氧化物阴极放电中的离子温度. , 2015, 64(18): 189401. doi: 10.7498/aps.64.189401
[5]	陈海军, 张耀文. 空间调制作用下Bessel型光晶格中物质波孤立子的稳定性. , 2014, 63(22): 220303. doi: 10.7498/aps.63.220303
[6]	石兰芳, 朱敏, 周先春, 汪维刚, 莫嘉琪. 一类非线性发展方程孤立子行波解. , 2014, 63(13): 130201. doi: 10.7498/aps.63.130201
[7]	张文玲, 马松华, 陈晶晶. (2+1)维Korteweg-de Vries方程的复合波解及局域激发. , 2014, 63(8): 080506. doi: 10.7498/aps.63.080506
[8]	陈海军, 李向富. 二维线性与非线性光晶格中物质波孤立子的稳定性. , 2013, 62(7): 070302. doi: 10.7498/aps.62.070302
[9]	王军民. 修正的Korteweg de Vries-正弦 Gordon方程的 Riemann 函数解. , 2012, 61(8): 080201. doi: 10.7498/aps.61.080201
[10]	套格图桑. Degasperis-Procesi 方程的无穷序列尖峰孤立波解. , 2011, 60(7): 070204. doi: 10.7498/aps.60.070204
[11]	石玉仁, 张娟, 杨红娟, 段文山. 耦合KdV方程的双峰孤立子及其稳定性. , 2011, 60(2): 020401. doi: 10.7498/aps.60.020401
[12]	吴红玉, 马松华, 方建平. (2+1)维 Korteweg-de Vries 方程的传播孤子及混沌行为. , 2010, 59(10): 6719-6724. doi: 10.7498/aps.59.6719
[13]	朱斌, 谷渝秋, 王玉晓, 刘红杰, 吴玉迟, 王磊, 王剑, 温贤伦, 焦春晔, 滕建, 何颖玲. 超短超强激光与稀薄等离子体相互作用中后孤立子的观测. , 2009, 58(2): 1100-1104. doi: 10.7498/aps.58.1100
[14]	高斌, 刘式达, 刘式适. Davey-Stewartson方程组的包络周期解和孤立波解. , 2009, 58(4): 2155-2158. doi: 10.7498/aps.58.2155
[15]	曹禹, 杨孔庆. 对声波和弹性波传播模拟的Hamilton系统方法. , 2003, 52(8): 1984-1992. doi: 10.7498/aps.52.1984
[16]	牛家胜, 马本堃. 在具有强非线性效应的离子晶体中光脉冲传播的孤立子特性. , 2002, 51(12): 2818-2822. doi: 10.7498/aps.51.2818
[17]	郑坚, 刘万东, 俞昌旋. 离子声波对电子输运的影响. , 2001, 50(4): 721-725. doi: 10.7498/aps.50.721
[18]	戴文龙, 贺贤土, 霍裕平, 刘之景. 等离子体中Langmuir波、横波和离子声波相互作用过程的孤立子行为. , 1987, 36(1): 67-73. doi: 10.7498/aps.36.67
[19]	贺贤土. 高频波激发低频磁场和离子声波的重整化强湍动理论. , 1986, 35(3): 283-299. doi: 10.7498/aps.35.283
[20]	罗正明. 环形轴对称系统准椭圆截面等离子体平衡的外部解. , 1978, 27(4): 448-458. doi: 10.7498/aps.27.448

计量

文章访问数: 6845
PDF下载量: 251
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板