Remarks on the preservation and breaking of translational symmetry for a class of ODEs
In this paper, we provide both a preservation and breaking of symmetry theorem for 2π-periodic problems of the form −u′′(t)+g(u(t))=f(t)u(0)−u(2π)=u′(0)−u′(2π)=0where g:R→Ris a given C1function and f:[0,2π]→Ris continuous. We provide a preservation of symmetry result that is analogous to one given b...
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| Format: | Article |
| Language: | English |
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Elsevier
2022-11-01
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| Series: | Examples and Counterexamples |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666657X22000167 |
| _version_ | 1811201498470678528 |
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| author | Edward Huynh Keoni Castellano |
| author_facet | Edward Huynh Keoni Castellano |
| author_sort | Edward Huynh |
| collection | DOAJ |
| description | In this paper, we provide both a preservation and breaking of symmetry theorem for 2π-periodic problems of the form −u′′(t)+g(u(t))=f(t)u(0)−u(2π)=u′(0)−u′(2π)=0where g:R→Ris a given C1function and f:[0,2π]→Ris continuous. We provide a preservation of symmetry result that is analogous to one given by (Willem, 1989) and a generalization of the theorem given by (Costa and Fang, 2019). Both of these theorems consider the group action of translation — which corresponds to periodicity. |
| first_indexed | 2024-04-12T02:22:40Z |
| format | Article |
| id | doaj.art-b9cdc2a334e54c7f9f6559435a865af9 |
| institution | Directory Open Access Journal |
| issn | 2666-657X |
| language | English |
| last_indexed | 2024-04-12T02:22:40Z |
| publishDate | 2022-11-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Examples and Counterexamples |
| spelling | doaj.art-b9cdc2a334e54c7f9f6559435a865af92022-12-22T03:52:05ZengElsevierExamples and Counterexamples2666-657X2022-11-012100079Remarks on the preservation and breaking of translational symmetry for a class of ODEsEdward Huynh0Keoni Castellano1Corresponding author.; Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Box 454020, NV, USADepartment of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Box 454020, NV, USAIn this paper, we provide both a preservation and breaking of symmetry theorem for 2π-periodic problems of the form −u′′(t)+g(u(t))=f(t)u(0)−u(2π)=u′(0)−u′(2π)=0where g:R→Ris a given C1function and f:[0,2π]→Ris continuous. We provide a preservation of symmetry result that is analogous to one given by (Willem, 1989) and a generalization of the theorem given by (Costa and Fang, 2019). Both of these theorems consider the group action of translation — which corresponds to periodicity.http://www.sciencedirect.com/science/article/pii/S2666657X22000167Periodic solutionsSymmetry breakingSymmetry preservationGroup actionCritical groupsMorse index |
| spellingShingle | Edward Huynh Keoni Castellano Remarks on the preservation and breaking of translational symmetry for a class of ODEs Examples and Counterexamples Periodic solutions Symmetry breaking Symmetry preservation Group action Critical groups Morse index |
| title | Remarks on the preservation and breaking of translational symmetry for a class of ODEs |
| title_full | Remarks on the preservation and breaking of translational symmetry for a class of ODEs |
| title_fullStr | Remarks on the preservation and breaking of translational symmetry for a class of ODEs |
| title_full_unstemmed | Remarks on the preservation and breaking of translational symmetry for a class of ODEs |
| title_short | Remarks on the preservation and breaking of translational symmetry for a class of ODEs |
| title_sort | remarks on the preservation and breaking of translational symmetry for a class of odes |
| topic | Periodic solutions Symmetry breaking Symmetry preservation Group action Critical groups Morse index |
| url | http://www.sciencedirect.com/science/article/pii/S2666657X22000167 |
| work_keys_str_mv | AT edwardhuynh remarksonthepreservationandbreakingoftranslationalsymmetryforaclassofodes AT keonicastellano remarksonthepreservationandbreakingoftranslationalsymmetryforaclassofodes |