A novel (3+1)-dimensional sine-Gorden and sinh-Gorden equation: Derivation, symmetries and conservation laws

Abstract In this paper, a novel (3+1)-dimensional sin-Gorden and sinh-Gorden eqaution 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 this equation can be reduced to the classical sin-Godon equation and the sinh-Godon equation. Some analytic solutions are presented by means of traveling wave transformation. Finally, based on the multiplier method, a conservation law is obtained.