Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions
| dc.contributor.area | Mathematics | en_US |
| dc.contributor.author | Mahmoudi, Fethi | en_US |
| dc.contributor.author | Malchiodi, Andrea | en_US |
| dc.contributor.author | Montenegro, Marcelo | en_US |
| dc.contributor.department | Functional Analysis and Applications | 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.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.identifier.uri | https://openscience.sissa.it/handle/1963/2112 | 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 |