On the asymptotic stability of bound states in 2D cubic Schroedinger equation
We consider the cubic nonlinear Schr\"{o}dinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space...
| Príomhchruthaitheoirí: | , |
|---|---|
| Formáid: | Journal article |
| Teanga: | English |
| Foilsithe / Cruthaithe: |
2006
|
| _version_ | 1826265298278285312 |
|---|---|
| author | Kirr, E Zarnescu, A |
| author_facet | Kirr, E Zarnescu, A |
| author_sort | Kirr, E |
| collection | OXFORD |
| description | We consider the cubic nonlinear Schr\"{o}dinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states. |
| first_indexed | 2024-03-06T20:21:29Z |
| format | Journal article |
| id | oxford-uuid:2dedd063-2a5c-4247-8e78-17bea628f85e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:21:29Z |
| publishDate | 2006 |
| record_format | dspace |
| spelling | oxford-uuid:2dedd063-2a5c-4247-8e78-17bea628f85e2022-03-26T12:46:01ZOn the asymptotic stability of bound states in 2D cubic Schroedinger equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2dedd063-2a5c-4247-8e78-17bea628f85eEnglishSymplectic Elements at Oxford2006Kirr, EZarnescu, AWe consider the cubic nonlinear Schr\"{o}dinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states. |
| spellingShingle | Kirr, E Zarnescu, A On the asymptotic stability of bound states in 2D cubic Schroedinger equation |
| title | On the asymptotic stability of bound states in 2D cubic Schroedinger
equation |
| title_full | On the asymptotic stability of bound states in 2D cubic Schroedinger
equation |
| title_fullStr | On the asymptotic stability of bound states in 2D cubic Schroedinger
equation |
| title_full_unstemmed | On the asymptotic stability of bound states in 2D cubic Schroedinger
equation |
| title_short | On the asymptotic stability of bound states in 2D cubic Schroedinger
equation |
| title_sort | on the asymptotic stability of bound states in 2d cubic schroedinger equation |
| work_keys_str_mv | AT kirre ontheasymptoticstabilityofboundstatesin2dcubicschroedingerequation AT zarnescua ontheasymptoticstabilityofboundstatesin2dcubicschroedingerequation |