双模耦合KdV方程的多孤子解与精确解

doi:10.3969/j.issn.1000-5641.2019.04.005

华东师范大学学报(自然科学版) ›› 2019, Vol. 2019 ›› Issue (4): 42-51.doi: 10.3969/j.issn.1000-5641.2019.04.005

双模耦合KdV方程的多孤子解与精确解

赵倩, 白喜瑞

宁波大学数学系, 浙江宁波 315211

收稿日期:2018-06-28 出版日期:2019-07-25 发布日期:2019-07-18
作者简介:赵倩,女,硕士研究生,研究方向为非线性数学物理.E-mail:15058429092@163.com;白喜瑞,女,硕士研究生,研究方向为偏微分方程.E-mail:15729216969@163.com.
基金资助:
国家自然科学基金（11471174）；宁波市自然科学基金（2014A610018）

Two-mode coupled KdV equation: Multiple-soliton solutions and other exact solutions

ZHAO Qian, BAI Xi-rui

Department of Mathematics, Ningbo University, Ningbo Zhejiang 315211, China

Received:2018-06-28 Online:2019-07-25 Published:2019-07-18

摘要/Abstract

摘要： 根据简化的Hirota双线性方法和Cole-Hopf变换，当一个新的双模耦合KdV方程中的非线性参数与耗散参数取特殊值时，得到了该新的双模耦合KdV方程的多孤子解.同时，当方程中的非线性参数与耗散参数取一般值时，通过不同的函数展开法，如tanh/coth法和Jacobi椭圆函数法，可得到这个方程的其他精确解.

关键词: 双模耦合KdV方程, 简化的Hirota方法, 多孤子解, 周期解

Abstract: In this paper, multiple-soliton solutions for a new two-mode coupled KdV (nTMcKdV) equation are obtained by using the simplified Hirota's method and the Cole-Hopf transformation. It is shown that these types of multiple solutions exist only for models in which specific values for the nonlinearity and dispersion parameters are included in the models. Furthermore, other exact solutions for an nTMcKdV equation using general values of these parameters are derived by using several different expansion methods such as the tanh/coth method and the Jacobi elliptic function method.

Key words: two-mode coupled KdV equation, simplified Hirota's method, multiplesoliton solutions, periodic solutions

中图分类号:

O178

赵倩, 白喜瑞. 双模耦合KdV方程的多孤子解与精确解[J]. 华东师范大学学报(自然科学版), 2019, 2019(4): 42-51.

ZHAO Qian, BAI Xi-rui. Two-mode coupled KdV equation: Multiple-soliton solutions and other exact solutions[J]. Journal of East China Normal University(Natural Sc, 2019, 2019(4): 42-51.

参考文献

[1] KORSUNSKY S V. Soliton solutions for a second-order KdV equation[J]. Phys Lett A, 1994, 185:174-176.
[2] LEE C T, LIU J L, LEE C C, et al. The second-order KdV equation and its soliton-like solution[J]. Modern Physics Letters B, 2009, 23:1771-1780.
[3] LEE C C, LEE C T, LIU J L, et al. Quasi-solitons of the two-mode Korteweg-de Vries equation[J]. Eur Phys J Appl Phys, 2010, 52:11301.
[4] LEE C T. Some notes on a two-mode Korteweg-de Vries equation[J]. Phys Scr, 2010, 81:065006.
[5] LEE C T, LIU J L. A Hamiltonian model and soliton phenomenon for a two-mode KdV equation[J]. Rocky Mt J Math, 2011, 41:1273-1289.
[6] LEE C T, LEE C C. On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system[J]. Waves in Random and Complex Media, 2013, 23:56-76.
[7] LEE C T, LEE C C. Analysis of solitonic phenomenon for a two-mode KdV equation[J]. Physics of Wave Phenomena, 2014, 22:69-80.
[8] LEE C T, LEE C C. On the study of a nonlinear higher order dispersive wave equation:Its mathematical physical structure and anomaly soliton phenomena[J]. Waves in Random and Complex Media, 2015, 25:197-222.
[9] LEE C T, LEE C C. Symbolic computation on a second-order KdV equation[J]. Journal of Symbolic Computation, 2016, 74:70-95.
[10] WAZWAZ A M. Multiple soliton solutions and other exact solutions for a two-mode KdV equation[J]. Math Methods Appl Sci, 2017, 40:2277-2283.
[11] LEE C T, LEE C C, LIU M L. Double-soliton and conservation law structures for a higher-order type of Korteweg-de Vries equation[J]. Physics Essays, 2015, 28:633-638.
[12] ALQURAN M, JARRAH A. Jacobi elliptic function solutions for a two-mode KdV equation[J/OL]. Journal of King Saud University-Science, (2017-07-03)[2018-06-28]. http://dx.doi.org/10.1016/j.jksus.2017.06.010.
[13] XIAO Z J, TIAN B, ZHEN H L, et al. Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid[J]. Waves in Random and Complex Media, 2017, 27:1-14.
[14] WAZWAZ A M. A two-mode modified KdV equation with multiple soliton solutions[J]. Appl Math Lett, 2017, 70:1-6.
[15] WAZWAZ A M. A two-mode Burgers equation of weak shock waves in a fluid:Multiple kink solutions and other exact solutions[J]. Int J Appl Comput Math, 2017, 3:3977-3985.
[16] WAZWAZ A M. A study on a two-wave mode Kadomtsev-Petviashvili equation:Conditions for multiple soliton solutions to exist[J]. Math Methods Appl Sci, 2017, 40:4128-4133.
[17] JARADAT H M, SYAM M, ALQURAN M. A two-mode coupled Korteweg-de Vries:Multiple-soliton solutions and other exact solutions[J]. Nonlinear Dyn, 2017, 90:371-377.
[18] WAZWAZ A M. Two-mode fifth-order KdV equations:Necessary conditions for multiple-soliton solutions to exist[J]. Nonlinear Dyn, 2017, 87:1685-1691.
[19] WAZWAZ A M. Two-mode Sharma-Tasso-Olver equation and two-mode fourth-order Burgers equation:Multiple kink solutions[J]. Alexandria Eng J, 2018, 57:1971-1976.
[20] JARDAT H M. Two-mode coupled Burgers equation:Multiple-kink solutions and other exact solutions[J]. Alexandria Eng J, 2018, 57:2151-2155.
[21] SYAM M, JARADAT H M, ALQURAN M. A study on the two-mode coupled modified Korteweg-de Vries using the simplified bilinear and the trigonometric-function methods[J]. Nonlinear Dyn, 2017, 90:1363-1371.
[22] WAZWAZ A M. Two wave mode higher-order modified KdV equations:Essential conditions for multiple soliton solutions to exist[J]. International Journal of Numerical Methods for Heat and Fluid Flow, 2017, 27:2223-2230.
[23] HEREMAN W, NUSEIR A. Symbolic methods to construct exact solutions of nonlinear partial differential equations[J]. Mathematics and Computers in Simulation, 1997, 43:13-27.
[24] WAZWAZ A M. Single and multiple-soliton solutions for the (2+1)-dimensional KdV equation[J]. Appl Math Comput, 2008, 204:20-26.
[25] ZUO J M, ZHANG Y M. The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq-Burgers equation[J]. Chin Phy B, 2011, 20:010205.
[26] WAZWAZ A M. Multiple soliton solutions for the integrable couplings of the KdV and the KP equations[J]. Open Physics, 2013, 11:291-295.
[27] WAZWAZ A M. Multiple kink solutions for two coupled integrable (2+1)-dimensional systems[J]. Appl Math Lett, 2016, 58:1-6.
[28] YU F J. Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy[J]. Chin Phys B, 2012, 21:010201.
[29] MALFLIET W, HEREMAN W. The tanh method:I. Exact solutions of nonlinear evolution and wave equations[J]. Phys Scr, 1996, 54:563-568.
[30] FAN E, HONA Y C. Generalized tanh method extended to special types of nonlinear equations[J]. Zeitschrift für Naturforschung A, 2002, 57:692-700.
[31] WAZWAZ A M. The tanh method for traveling wave solutions of nonlinear equations[J]. Appl Math and Comput, 2004, 154:713-723.
[32] LIU S, FU Z, LIU S, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Phys Lett A, 2001, 289:69-74.

双模耦合KdV方程的多孤子解与精确解

Two-mode coupled KdV equation: Multiple-soliton solutions and other exact solutions

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