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

$$\{\begin{array}{rlr}& u_{tt}-u_{xx}+u+\alpha uv+\beta |u{|}^{2}u=0,& v_{tt}-v_{xx}=(|u{|}^2{)}_{xx},\end{array}$$

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 Lam\'{e} equation and Floquet theory. When period $L\to \mathrm{\infty }$, 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.