Complex dynamics in one-dimensional nonlinear Schrödinger equations with stepwise potential
type: Item
instance of: scholarly article
title: Complex dynamics in one-dimensional nonlinear Schrödinger equations with stepwise potential
zbMATH Open document ID: 1407.35190
DOI: 10.1155/2018/2101482
author: Chiara Zanini
author: Fabio Zanolin
published in: Complexity
publication date: 2019-02-19T00:00:00Z
review text: Summary: We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation \(-\varepsilon^2 u''+V(x)u=f(u),\) 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 \(\varepsilon >0\), although small, can be explicitly estimated.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 7025140
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/2101482
OpenAlex ID: W2902982411
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