Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

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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Main Authors: Chiara Zanini, Fabio Zanolin
Format: Article
Language:English
Published: Hindawi-Wiley 2018-01-01
Series:Complexity
Online Access:http://dx.doi.org/10.1155/2018/2101482
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spelling doaj-32da4eb8dd194d17a333d70a870fa2c62020-11-24T21:05:17ZengHindawi-WileyComplexity1076-27871099-05262018-01-01201810.1155/2018/21014822101482Complex 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
collection DOAJ
language English
format Article
sources DOAJ
author Chiara Zanini
Fabio Zanolin
spellingShingle Chiara Zanini
Fabio Zanolin
Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential
Complexity
author_facet Chiara Zanini
Fabio Zanolin
author_sort Chiara Zanini
title 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_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_sort complex dynamics in one-dimensional nonlinear schrödinger equations with stepwise potential
publisher Hindawi-Wiley
series Complexity
issn 1076-2787
1099-0526
publishDate 2018-01-01
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.
url http://dx.doi.org/10.1155/2018/2101482
work_keys_str_mv AT chiarazanini complexdynamicsinonedimensionalnonlinearschrodingerequationswithstepwisepotential
AT fabiozanolin complexdynamicsinonedimensionalnonlinearschrodingerequationswithstepwisepotential
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