一类高阶KdV方程的行波复化亚纯解  

The Traveling Wave Meromorphic Solutions of a Kind of Complex Higher-Order KdV Equations

作　　者：袁文俊[1] 党国强[1] YUAN Wenjun;DANG Guoqiang(School of Mathematics and Information Science, Guangzhou University, Guangzhou Guangdong 510006, China)

机构地区：[1]广州大学数学与信息科学学院,广东广州510006

出　　处：《江西师范大学学报（自然科学版）》2017年第6期579-584,共6页Journal of Jiangxi Normal University(Natural Science Edition)

基　　金：国家自然科学基金(11271090);广东省自然科学基金(2016A030310257;2015A030313346)资助项目

摘　　要：考虑了一类高阶KdV微分方程u_t+δu^2u_x+βu_xu_(xx)+γuu_(xxx)+ωu_(xxxxx)=0.通过行波变换u(x,t)=w(z),z=x+λt(λ≠0),这类高阶KdV微分方程变为常微分方程w^(4)+δww″+βw'2+γw^3+λw+μ=0,其控制项有4项:E(z,w)=w(4)+δww″+βw'2+γw3.主要结果是运用复方法给出这些常微分方程的3类亚纯解表达式,即椭圆函数解、有理函数解、eαz(α∈C)的有理函数解,并以行波复化modified Sawada-Kotera方程u_t+u_(xxxxx)+5uu_(xxx)+15u_xu_(xx)+5u^2u_x=0,Kaup-Kupershmid方程u_t-u_(xxxxx)+20uu_(xxx)+50u_xu_(xx)-80u^2u_x=0为例说明:除了该文所确定的亚纯解之外,或许有方程还有其他的亚纯解.In this paper,a class of fifth-order KdV equations u t+δu2u x+βu xu xx+γuu xxx+ωu xxxxx=0are considered.These KdV equations are reduced as a class of complex ordinary differential equations w(4)+δww″+βw′2+γw3+λw+μ=0,by using the traveling wave transformation u(x,t)=w(z),z=x+λt(λ≠0),whose dominant parts have four terms E∧(z,w)=w(4)+δww″+βw′2+γw3.The main result is that the meromorphic solutions of the equations are obtained as elliptic function solutions,rational function solutions,and rational function solutions of eαz(α∈C),by employing complex method.Furthermore,two examples u t+u xxxxx+5uu xxx+15u xu xx+5u2u x=0(modified Sawada-Kotera equation)and u t-u xxxxx+20uu xxx+50u xu xx-80u2u x=0(Kaup-Kupershmid equation)are given to show that besides the meromorphic solutions which are confirmed,some equations maybe have other meromorphic solutions.

关 键 词：高阶KDV方程 微分方程 亚纯函数 椭圆函数 

分 类 号：O174.52[理学—数学]