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Article overview
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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 | | Fethi Mahmoudi
; Andrea Malchiodi
; Marcelo Montenegro
; | | Date: |
1 Aug 2007 | | 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. | | Source: | arXiv, 0708.0125 | | Services: | Forum | Review | PDF | Favorites | |
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