摘要
以(G'/G)的基本思想为依据,构造了一种变系数G展开法,即(G-G'/G+G')展开法,其中的函数G满足一类二阶变系数非线性常微分方程.通过此展开法,并借助Mathematica计算软件,对广义浅水波方程进行了求解,获得了该方程显式行波解.事实证明,变系数G展开法对于求解非线性偏微分方程的精确解是有效可行的.
In this paper,based on the basic idea of the(G'/G) expansion method,a class variable coefficient G expansion method is constructed,that is the (G-G'/G+G')expansion method,which satisfies a class of two order nonlinear ordinary differential equations with variable coefficients. Through this expansion method,the generalized shallow water wave equation is solved with the help of the calculation software Mathematica,and the explicit traveling wave solutions of the equation are obtained. It is proved that the variable coefficient G expansion method is effective and feasible for solving the exact solutions of the nonlinear partial differential equations.
引文
[1]WANG M L,LI X Z,ZHANG J L.The(G'/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J].Physics Letters A,2008,372(4):417-423.
[2]WANG M L,ZHANG J L,LI X Z.Application of the(G'/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations[J].Applied Mathematics and Computation,2008,206(1):321-326.
[3]LI L X,WANG M L.The(G'/G)-expansion method and travelling wave solutions for a higher-order nonlinear schrodinger equation[J].Applied Mathematics and Computation,2009,208(2):440-445.
[4]王鑫.一类非线性偏微分方程的精确解[J].应用数学,2013,26(3):521-525.
[5]曹瑞.一类广义Zakharov方程的精确行波解[J].数学杂志,2013,33(5):837-843.
[6]WHITHAM G B.Linear and nonlinear waves[M].New York:Wiley-Interscience,1974.
[7]CLARKSON P A,MANSFIELD E L.On a shallow water wave equation[J].Nonlinearity,1994,7(3):975-1000.
[8]HIETARINTA J.Hirota’s bilinear method and partial integrability[M].Berlin:Springer,1990:459-478.
[9]沈守枫.(1+1)维广义的浅水波方程的变量分离解和孤子激发模式[J].物理学报,2006,55(3):1016-1022.
[10]王鑫,邢文雅,李胜军.广义浅水波方程新的行波解[J].大学数学,2015,31(4):9-13.