(n+1)维Sinh-Gordon方程新的椭圆函数周期解

New solutions of Jacobian elliptic and periodical function to (n+1)-dimensional Sinh-Gordon equation

通过引入一个函数变换将(n+1)维Sinh-Gordon方程转化为新的多项式型的非线性偏微分方程.然后由行波约化将其常微分方程化,在拟设法、齐次平衡法和Jacobi椭圆函数法的基础上,借助Mathematica软件和新近提出的F-展开法,求出并研究了(n+1)维SG方程的Jacobi椭圆函数表示的双周期波解,分析了解的结构,在极限情况下这些解退化为相应的孤立波解、三角函数解和奇异行波解.利用数学软件绘出了对应的图形.为进一步研究(n+1)维SG方程在众多的自然科学领域的更广泛的应用提供了理论依据.

By a transformation of logarithm function, the （ n ＋ 1 ） -dimensional Sinh-Gordon equation is converted into a new polynomial type of nonlinear partial differential equation. Then it is reduced to an ordinary differential equation by the wave transformation. By using Mathematica and the F - expansion method recently proposed on the base of analogic method, homogeneous balance method and Jacobi method, the double periodic wave solutions expressed by Jacobi elliptic functions for the （ n ＋ 1 ） -dimensional Sinh-Gordon equation are drived. In the limit cases, the solitary wave solutions, triangle function solutions and some type of traveling wave solutions for the system are obtained, and the structures of these solutions are analysed. Finally some corresponding pictures drawn by Mathematica are provided. The results provide theoretical references for further studies in this field.