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# Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions

 dc.contributor.author Mahmoudi, Fethi en_US dc.contributor.author Malchiodi, Andrea en_US dc.contributor.author Montenegro, Marcelo en_US dc.date.accessioned 2007-09-17T09:14:27Z en_US dc.date.accessioned 2011-09-07T20:28:05Z dc.date.available 2007-09-17T09:14:27Z en_US dc.date.available 2011-09-07T20:28:05Z dc.date.issued 2007-09-17T09:14:27Z en_US dc.identifier.uri http://preprints.sissa.it/xmlui/handle/1963/2112 en_US dc.description.abstract We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\epsilon$ a small parameter corresponding to the Planck constant. As $\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\epsilon$. Based on these, an existence result will be proved in the second part. en_US dc.format.extent 498331 bytes en_US dc.format.mimetype application/pdf en_US dc.language.iso en_US en_US dc.relation.ispartofseries SISSA;51/2007/M en_US dc.relation.ispartofseries arXiv.org;0708.0125 en_US dc.title Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions en_US dc.type Preprint en_US dc.contributor.department Functional Analysis and Applications en_US dc.contributor.area Mathematics en_US
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