On periodic waves for sine- and sinh-Gordon equations

Results of existence as well as orbital stability/instability of periodic waves associated with sine-Gordon and sinh-Gordon equations will be treated in this manuscript. The sine-Gordon equation is known to be completely integrable, exactly solvable and it has several special solutions as soliton and breather solutions. We obtain in both cases, a periodic sign-changed wave which is smooth when posed on whole real line. The main tool used is the Grillakis, Shatah and Straussʼ theory.