(n+1)维Sine-Gordon方程的一类精确解

A class of exact solutions to the(n+1) dimensional Sine-Gordon equation

在描述Sine-Gordon方程对应的背景问题时,非线状精确解往往比行波解更准确更深刻。为了得到两类(n+1)维Sine-Gordon方程的非线状精确解,该文先利用拟设法和变量分离法求出(1+1)维Sine-Gordon方程的一类非线状精确解,再利用线性变换把两类高维Sine-Gordon方程转换成(1+1)维Sine-Gordon方程,从而得到了高维Sine-Gordon方程的呼吸孤子解和钟形孤子解等一类精确解,这些新精确解都具有代表性。

Nonlinear exact solutions to the Sine-Gordon equation are always more exact and profound than travelling solutions when describing its related backgrounds. In order to obtain nonlinear exact solutions to two classes of （re＋1） dimensional Sine-Gordon equations, a class of nonlinear exact solutions to （1＋1） dimensional Sine-Gordon equation was first found out by the ansatz method and variable separation method. Then, under suitable linear transformations, the high dimensional classical/generalized Sine-Gordon equations were reduced to the （1 ＋ 1） dimensional one, and so a sort of solutions including the breather-soliton and the bell-soliton ones to these high dimensional equations was obtained. These solutions obtained here are all typical.