Orbital stability of periodic standing waves of the coupled Klein-Gordon-Zakharov equations

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations {u(tt) - u(xx) + u + alpha uv +beta|u|(2)u = 0, v(tt) - v(xx) = (|u|(2))(xx), where alpha > 0 and beta are two real numbers and alpha > beta. Under some suitable conditions, we show the existence of a smooth curve positive standing wave solutions of dnoidal type with a fixed fundamental period L for the above equations. Further, we obtain the stability of the dnoidal waves for the coupled Klein-Gordon-Zakharov equations by applying the abstract stability theory and combining the detailed spectral analysis given by using Lame ' equation and Floquet theory. When period L -> infinity, dnoidal type will turn into sech-type in the sense of limit. In such case, we can obtain stability of sech-type standing waves. In particular, beta = 0 is advisable, we still can show the the stability of the dnoidal type and sech-type standing waves for the classical Klein-Gordon-Zakharov equations.