Orbital stability for periodic standing waves of the Klein-Gordon-Zakharov system and the beam equation

The existence and stability of spatially periodic waves in the Klein-Gordon-Zakharov (KGZ) system are studied. We show a local existence result for low regularity initial data. Then, we construct a one-parameter family of periodic dnoidal waves for (KGZ) system when the period is bigger than . We show that these waves are stable whenever an appropriate function satisfies the standard Grillakis-Shatah-Strauss (Grillakis et al. J Funct Anal 74(1):160-197, 1987; Grillakis et al. J Funct Anal 94(2):308-348, 1990) type condition. We compute the intervals for the parameter omega explicitly in terms of L and by taking the limit L -> a we recover the previously known stability results for the solitary waves in the whole line case. For the beam equation, we show the existence of spatially periodic standing waves and show that orbital stability holds if an appropriate functional satisfies Grillakis-Shatah-Strauss type condition.