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

Abstract

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations \begin{document} $ \begin{equation*} \left\{ \begin{aligned} &u_{tt}-u_{xx}+u+\alpha uv+\beta|u|^{2}u = 0, \ &v_{tt}-v_{xx} = (|u|^{2})_{xx}, \end{aligned} \right. \end{equation*} $ \end{document} 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 $L\rightarrow\infty$, 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.