@article{Li2023, 
author = {Qiuying Li and Xiaoxiao Zheng and Zhenguo Wang},
title = {Orbital stability of periodic standing waves of the coupled Klein-Gordon-Zakharov equations},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {4},
pages = {8560-8579},
keywords = {coupled Klein-Gordon-Zakharov equations, periodic standing waves, orbital stability, Floquet theory, Hamiltonian system},
url = {https://www.sciopen.com/article/10.3934/math.2023430},
doi = {10.3934/math.2023430},
abstract = {This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations               {                                                u                          t              t                                −                      u                          x              x                                +          u          +          α          u          v          +          β                      |                    u                                    |                                      2                                u          =          0          ,                                                v                          t              t                                −                      v                          x              x                                =          (                      |                    u                                    |                                      2                                            )                          x              x                                ,                        where    α  >  0 and    β are two real numbers and    α  >  β. 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  →  ∞, 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,    β  =  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.}
}