Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential
We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinea...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2018/2101482 |
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| author | Chiara Zanini Fabio Zanolin |
| author_facet | Chiara Zanini Fabio Zanolin |
| author_sort | Chiara Zanini |
| collection | DOAJ |
| description | We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated. |
| format | Article |
| id | doaj-art-32da4eb8dd194d17a333d70a870fa2c6 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-32da4eb8dd194d17a333d70a870fa2c62025-02-03T01:10:56ZengWileyComplexity1076-27871099-05262018-01-01201810.1155/2018/21014822101482Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise PotentialChiara Zanini0Fabio Zanolin1Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, ItalyUniversità di Udine, Dipartimento di Science Matematiche, Informatiche e Fisiche, Via delle Scienze 206, 33100 Udine, ItalyWe prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.http://dx.doi.org/10.1155/2018/2101482 |
| spellingShingle | Chiara Zanini Fabio Zanolin Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential Complexity |
| title | Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential |
| title_full | Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential |
| title_fullStr | Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential |
| title_full_unstemmed | Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential |
| title_short | Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential |
| title_sort | complex dynamics in one dimensional nonlinear schrodinger equations with stepwise potential |
| url | http://dx.doi.org/10.1155/2018/2101482 |
| work_keys_str_mv | AT chiarazanini complexdynamicsinonedimensionalnonlinearschrodingerequationswithstepwisepotential AT fabiozanolin complexdynamicsinonedimensionalnonlinearschrodingerequationswithstepwisepotential |