In this paper, a novel (3+1)-dimensional sine-Gorden and a sinh-Gorden equation are derived. These two equations are derived for the first time from the extended (3+1) dimensional zero curvature equation, using the compatibility condition. Then the infinitesimal transformation of this equation is studied from the symmetry point of view. Meanwhile, it turns out that these two equations can be reduced to the classical sin-Gordon equation and the sinh-Gordon equation. Some analytic solutions are presented by means of traveling wave transformation. Finally, based on the multiplier method, a conservation law is obtained.

中文翻译：

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一个新颖的（3 + 1）维正弦-Gorden和sinh-Gorden方程：导数，对称性和守恒律

本文推导了新颖的（3 + 1）维正弦-Gorden方程和sinh-Gorden方程。这两个方程式是首次使用相容性条件从扩展的（3 + 1）维零曲率方程式导出的。然后从对称性的角度研究该方程的无穷小变换。同时，事实证明，这两个方程可以简化为经典的sin-Gordon方程和sinh-Gordon方程。通过行波变换给出了一些解析解。最后，基于乘数法，得到了一个守恒律。