用修正的F-展开法求解(n+1)维Sine-Gordon方程

Solution of (n+1)-dimensional Sine-Gordon equation with modified F-expansion method

用一个未知函数的变换将(n+1)维Sine-Gordon方程转化为新未知函数及其偏导数为变元的多项式型的非线性偏微分方程.在拟设法、齐次平衡法和Jacobi椭圆函数法的基础上,借助Mathematica软件和修正的F-展开法,求出了(n+1)维SG方程的Weierstrass椭圆函数解、Jacobi椭圆函数表示的双周期波解,研究了极限情况下解的退化形式,利用数学软件绘出了部分解对应的图形.研究表明,许多解在欧氏变换下是等价的.

By means of a transformation of unknown function, the （n＋1）-dimensional Sine-Gordon equation was converted into a nonlinear partial differential equation of a polynomial type with a new unknown function and its partial derivatives. Using the software Mathematica and modified F-expansion method, the Weierstrass elliptic functions solutions, double periodic wave solutions expressed by Jacobi elliptic functions for the （n ＋1）-dimensional Sine-Gordon equation were obtained, on the base of analogic method, homogeneous balance method and Jacobi method. In the limit cases, the degenerate solutions were researched and their some corresponding graphics drawn with Mathematica were given. It was shown by the study that many of the solutions were equivalent for Euclid transformation.