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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (8): 81-89.doi: 10.6040/j.issn.1671-9352.0.2018.706

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一类辅助微分方程的亚纯解及其应用

古勇毅,孔荫莹*   

  1. 广东财经大学统计与数学学院, 广东 广州 510320
  • 出版日期:2019-08-20 发布日期:2019-07-03
  • 作者简介:古勇毅(1985— ),男,博士,讲师,研究方向为复分析及其应用. E-mail:gdguyongyi@163.com*通信作者简介:孔荫莹(1979— ),男,博士,教授,研究方向为复分析及其应用. E-mail:kongcoco@hotmail.com
  • 基金资助:
    广东省自然科学基金基金资助项目(2018A030313954);广东省普通高校基础研究重大项目(2017KZDXM038);广州市社会科学界联合会2018年度“羊城青年学人”资助项目(18QNXR35)

Meromorphic solutions of a class of auxiliary differential equation and its applications

GU Yong-yi, KONG Yin-ying*   

  1. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, Guangdong, China
  • Online:2019-08-20 Published:2019-07-03

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摘要： 介绍了寻求非线性偏微分方程精确解的方法——复方法,用该方法研究了一类辅助微分方程的亚纯解,并将所得结果运用于寻求相关的非线性偏微分方程的精确解,得到Vakhnenko-Parkes方程和Dodd-Bullough-Mikhailov方程的精确解。

关键词: 微分方程, 亚纯解, Vakhnenko-Parkes方程, Dodd-Bullough-Mikhailov方程

Abstract: This paper introduce a method to find exact solutions of nonlinear partial differential equations—complex method, and derive meromorphic solutions for a class of algebraic differential equation by the mentioned method. The results are used to seek exact solutions of nonlinear differential equations. Exact solutions of the Vakhnenko-Parkes equation and Dodd-Bullough-Mikhailov equation are obtained.

Key words: differential equation, meromorphic solution, Vakhnenko-Parkes equation, Dodd-Bullough-Mikhailov equation

中图分类号: 

  • O174.52
[1] EREMENKO A. Meromorphic solutions of equations of Briot-Bouquet type[J]. Teorija Funktsii Funktsional Analiz i Prilozhen, 1982, 38:48-56.
[2] EREMENKO A, LIAO Liangwen, NG Tuenwai. Meromorphic solutions of higher order Briot-Bouquet differential equations[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 2009, 146(1):197-206.
[3] EREMENKO A. Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation[J]. Journal of Mathematical Physics Analysis Geometry, 2006, 2(3):278-286.
[4] KUDRYASHOV N A. Meromorphic solutions of nonlinear ordinary differential equations[J]. Communications in Nonlinear Sciences and Numerical Simulations, 2010, 15(10):2778-2790.
[5] DEMINA M V, KUDRYASHOV N A. From Laurent series to exact meromorphic solutions: the Kawahara equation[J]. Physics Letters A, 2010, 374(39):4023-4029.
[6] KUDRYASHOV N A, SINELSHCHIKOV D I, DEMINA M V. Exact solutions of the generalized Bretherton equation[J]. Physics Letters A, 2011, 375(7):1074-1079.
[7] YUAN Wenjun, SHANG Yadong, HUANG Yong, et al. The representation of meromorphic solutions to certain ordinary differential equations and its applications[J]. Science China Mathematics, 2013, 43(6):563-575.
[8] YUAN Wenjun, LI Yezhou, LIN Jianming. Meromorphic solutions of an auxiliary ordinary differential equation using complex method[J]. Mathematical Methods in the Applied Sciences, 2013, 36(13):1776-1782.
[9] YUAN Wenjun, WU Yonghong, CHEN Qiuhui, et al. All meromorphic solutions for two forms of odd order algebraic differential equations and its applications[J]. Applied Mathematics and Computation, 2014, 240:240-251.
[10] YUAN Wenjun, XIONG Weiling, LIN Jianming, et al. All meromorphic solutions of an auxiliary ordinary differential equation and its applications[J]. Acta Mathematica Scientia, 2015, 35(5):1241-1250.
[11] LANG S. Elliptic functions[M]. New York: Springer, 1987.
[12] CONTE R, MUSETTE M. Elliptic general analytic solutions[J]. Studies in Applied Mathematics, 2009, 123(1):63-81.
[13] VAKHNENKO V O. Solitons in a nonlinear model medium[J]. Journal of Physics A: General Physics, 1992, 25(15):4181-4187.
[14] VAKHNENKO V O, PARKES E J. The two loop soliton solution of the Vakhnenko equation[J]. Nonlinearity, 1998, 11(6):1457-1464.
[15] DODD R K, BULLOUGH R K. Polynomial conserved densities for the sine-Gordon equations[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1977, 352(1671):481-503.
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