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

Qiuying Li; Xiaoxiao Zheng; Zhenguo Wang; Qiuying Li; Xiaoxiao Zheng; Zhenguo Wang

doi:10.3934/math.2023430

AIMS Mathematics

2023, Volume 8, Issue 4: 8560-8579. doi: 10.3934/math.2023430

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Research article

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

1.
School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu 273155, China
3.
Department of Mathematics, Taiyuan University, Taiyuan 030032, China

Received: 03 October 2022 Revised: 19 January 2023 Accepted: 27 January 2023 Published: 06 February 2023
MSC : 35B35, 35C08, 35R10

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations

$ \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*} $

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.
- coupled Klein-Gordon-Zakharov equations,
- periodic standing waves,
- orbital stability,
- Floquet theory,
- Hamiltonian system
Citation: Qiuying Li, Xiaoxiao Zheng, Zhenguo Wang. Orbital stability of periodic standing waves of the coupled Klein-Gordon-Zakharov equations[J]. AIMS Mathematics, 2023, 8(4): 8560-8579. doi: 10.3934/math.2023430

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Abstract

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations

$ \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*} $

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.

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