JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (8): 81-89.doi: 10.6040/j.issn.1671-9352.0.2018.706

Previous Articles     Next Articles

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

PDF (PC)

620

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

CLC Number: 

  • 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.
[1] ZHANG Ru, HAN Xu, LIU Xiao-gang. Convergence and contractivity of boundary value methods for nonlinear delay differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 97-101.
[2] ZHANG Huan, LI Yong-xiang. Positive periodic solutions of higher-order ordinary differential equations with delayed derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 29-36.
[3] CHEN Yu-jia, YANG He. Existence of periodic solutions of a class of third order delay differential equations in Banach spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(8): 84-94.
[4] . Further results on the growth of solutions of linear complex differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 23-29.
[5] XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.
[6] CHEN Li, LIN Ling. Stock option pricing with time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 36-41.
[7] YANG Xu, LI Shuo. Comparison theorem for backward doubly stochastic differential equations driven by white noises and Poisson random measures [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 26-29.
[8] LI Tao-tao. Existence of radial positive solutions of second-order semi-positone elliptic differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 48-55.
[9] ZHANG Sha, JIA Mei, LI Yan, LI Xiao-chen. Existence and uniqueness of solutions for three point boundary value problems of impulsive fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 66-72.
[10] ZHANG Di, LIU Wen-bin. Existence of solutions for p(t)-Laplacian fractional infinite-point boundary value problems at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 72-80.
[11] FENG Hai-xing, ZHAI Cheng-bo. Multiple positive solutions of a system of high order nonlinear fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 48-57.
[12] CUI Jing, LIANG Qiu-ju. Existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 81-88.
[13] . Application of fuzzy differential transform method for solving fuzzy integral differential equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 42-49.
[14] SU Xiao-feng, JIA Mei, LI Meng-meng. Existence of solution for fractional differential equation integral boundary value problem at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 66-73.
[15] HU Pei-chu, WU Ai-di. Topics in L-functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 1-15.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] YANG Lun, XU Zheng-gang, WANG Hui*, CHEN Qi-mei, CHEN Wei, HU Yan-xia, SHI Yuan, ZHU Hong-lei, ZENG Yong-qing*. Silence of PID1 gene expression using RNA interference in C2C12 cell line[J]. J4, 2013, 48(1): 36 -42 .
[2] LIU Yan-ping, WU Qun-ying. Almost sure limit theorems for the maximum of Gaussian sequences#br# with optimized weight[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(05): 50 -53 .
[3] ZHANG Shen-gui. Multiplicity of solutions for local superlinear p-kirchhoff-type equation#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(05): 61 -68 .
[4] Ming-Chit Liu. THE TWO GOLDBACH CONJECTURES[J]. J4, 2013, 48(2): 1 -14 .
[5] WANG Kai-rong, GAO Pei-ting. Two mixed conjugate gradient methods based on DY[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 16 -23 .
[6] LI Ya-nan1, LIU Lei-po2, WANG Yu-guang3. Passive sliding mode control for uncertain time-delay systems subjected to input nonlinearity[J]. J4, 2010, 45(6): 99 -104 .
[7] XU Guang-zhu1, LIU Ming2, REN Dong1, MA Yi-de3, LIU Xiao-li1. Multi-region image segmentation based on pulse coupled neural network[J]. J4, 2010, 45(7): 86 -93 .
[8] QU Xiao-ying ,ZHAO Jing . Solution of the Klein-Gordon equation for the time-dependent potential[J]. J4, 2007, 42(7): 22 -26 .
[9] WANG Qi,ZHAO Hong-luan . [J]. J4, 2006, 41(6): 84 -86 .
[10] JIN Li-ming,YANG Yan*,LIU Wan-shun,HAN Bao-qin,TIAN Wen-jie,FAN Sheng-di . Protective effects of chitosan oligosaccharide and its derivatives on carbon tetrachloride-induced liver damage in mice[J]. J4, 2007, 42(7): 1 -04 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd