Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions
| dc.creator | Mahmoudi, Fethi | |
| dc.creator | Malchiodi, Andrea | |
| dc.creator | Montenegro, Marcelo | |
| dc.date | 2007-08-01 | |
| dc.date.accessioned | 2026-07-07T08:21:43Z | |
| dc.date.available | 2026-07-07T08:21:43Z | |
| dc.description | We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ 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 $ε^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $ε$. Based on these, an existence result will be proved in the second part. | |
| dc.description | 44 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0708.0125 | |
| dc.identifier | http://arxiv.org/abs/0708.0125 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/135409 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 34B18, 35B25, 35B34, 35J20, 35J60 | |
| dc.title | Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions | |
| dc.type | text |