New solitary and double periodic wave solutions for a generalized sinh-Gordon equation

Our methodology is based on introducing the binary Kudryashov method for finding double periodic wave solutions to the generalized sinh-Gordon equation. This method is a powerful tool for searching the exact traveling solutions of nonlinear partial differential equations in mathematical physics. These exact solutions demonstrate the fact that solutions to the Generalized sinh-Gordon equation can exhibit a variety of behaviors. The solutions are constructed elegantly, without the need for complicated analytical algorithms.