{"title":"耦合Klein-Gordon-Zakharov方程周期驻波的轨道稳定性","authors":"Qiuying Li, Xiaoxiao Zheng, Zhenguo Wang","doi":"10.3934/math.2023430","DOIUrl":null,"url":null,"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.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Orbital stability of periodic standing waves of the coupled Klein-Gordon-Zakharov equations\",\"authors\":\"Qiuying Li, Xiaoxiao Zheng, Zhenguo Wang\",\"doi\":\"10.3934/math.2023430\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":48562,\"journal\":{\"name\":\"AIMS Mathematics\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"AIMS Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/math.2023430\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"AIMS Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/math.2023430","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
摘要
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 Lamé 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.
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{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.
期刊介绍:
AIMS Mathematics is an international Open Access journal devoted to publishing peer-reviewed, high quality, original papers in all fields of mathematics. We publish the following article types: original research articles, reviews, editorials, letters, and conference reports.