Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation: utt − uxx + u − |u|2u = 0. We also present a result of orbital instability of snoidal standing wave solutions to the Klein–Gordon equation utt − uxx + |u|2u = 0. The main tool to obtain these results is the classical Grillakis, Shatah and Strauss’ theory in the periodic context.