Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential
Abstract In this paper, the nonlinear wave equation with singular Legendre potential utt−uxx+VL(x)u+mu+perturbation=0 $$ u_{tt}-u_{xx}+V_{L}(x)u + mu + \mathit{perturbation}=0 $$ subject to certain boundary conditions is considered, where m is a positive real number and VL(x)=−12−14tan2x $V_{L}(x)=-...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-06-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-019-1225-x |
| _version_ | 1819015745123647488 |
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| author | Guanghua Shi Dongfeng Yan |
| author_facet | Guanghua Shi Dongfeng Yan |
| author_sort | Guanghua Shi |
| collection | DOAJ |
| description | Abstract In this paper, the nonlinear wave equation with singular Legendre potential utt−uxx+VL(x)u+mu+perturbation=0 $$ u_{tt}-u_{xx}+V_{L}(x)u + mu + \mathit{perturbation}=0 $$ subject to certain boundary conditions is considered, where m is a positive real number and VL(x)=−12−14tan2x $V_{L}(x)=-\frac{1}{2}-\frac{1}{4}\tan ^{2} {x}$, x∈(−π2,π2) $x\in (-\frac{\pi }{2},\frac{\pi }{2})$. By means of the partial Birkhoff normal form technique and infinite-dimensional Kolmogorov–Arnold–Moser theory, it is proved that, for every m∈R+∖{14} $m\in \mathbb{R}_{+}\setminus \{\frac{1}{4}\}$, the above equation admits plenty of quasi-periodic solutions with three frequencies. |
| first_indexed | 2024-12-21T02:36:37Z |
| format | Article |
| id | doaj.art-d2d382bc973e4743a05f6285c1b6b7ee |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-21T02:36:37Z |
| publishDate | 2019-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-d2d382bc973e4743a05f6285c1b6b7ee2022-12-21T19:18:47ZengSpringerOpenBoundary Value Problems1687-27702019-06-012019111810.1186/s13661-019-1225-xQuasi-periodic solutions for nonlinear wave equation with singular Legendre potentialGuanghua Shi0Dongfeng Yan1College of Mathematics and Computer Science, Hunan Normal UniversitySchool of Mathematics and Statistics, Zhengzhou UniversityAbstract In this paper, the nonlinear wave equation with singular Legendre potential utt−uxx+VL(x)u+mu+perturbation=0 $$ u_{tt}-u_{xx}+V_{L}(x)u + mu + \mathit{perturbation}=0 $$ subject to certain boundary conditions is considered, where m is a positive real number and VL(x)=−12−14tan2x $V_{L}(x)=-\frac{1}{2}-\frac{1}{4}\tan ^{2} {x}$, x∈(−π2,π2) $x\in (-\frac{\pi }{2},\frac{\pi }{2})$. By means of the partial Birkhoff normal form technique and infinite-dimensional Kolmogorov–Arnold–Moser theory, it is proved that, for every m∈R+∖{14} $m\in \mathbb{R}_{+}\setminus \{\frac{1}{4}\}$, the above equation admits plenty of quasi-periodic solutions with three frequencies.http://link.springer.com/article/10.1186/s13661-019-1225-xKolmogorov–Arnold–Moser theoryQuasi-periodic solutionsSingular differential operatorBirkhoff normal form |
| spellingShingle | Guanghua Shi Dongfeng Yan Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential Boundary Value Problems Kolmogorov–Arnold–Moser theory Quasi-periodic solutions Singular differential operator Birkhoff normal form |
| title | Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential |
| title_full | Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential |
| title_fullStr | Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential |
| title_full_unstemmed | Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential |
| title_short | Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential |
| title_sort | quasi periodic solutions for nonlinear wave equation with singular legendre potential |
| topic | Kolmogorov–Arnold–Moser theory Quasi-periodic solutions Singular differential operator Birkhoff normal form |
| url | http://link.springer.com/article/10.1186/s13661-019-1225-x |
| work_keys_str_mv | AT guanghuashi quasiperiodicsolutionsfornonlinearwaveequationwithsingularlegendrepotential AT dongfengyan quasiperiodicsolutionsfornonlinearwaveequationwithsingularlegendrepotential |