Uniform semiclassical approximations of the nonlinear Schroedinger equation by a Painleve mapping
| Autor: | D. Witthaut, H. J. Korsch |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2006 |
| Předmět: |
Physics
Condensed Matter::Quantum Gases Quantum Physics Differential equation FOS: Physical sciences General Physics and Astronomy Semiclassical physics Statistical and Nonlinear Physics Eigenfunction Schrödinger equation Quantization (physics) symbols.namesake Ordinary differential equation symbols Quantum Physics (quant-ph) Wave function Nonlinear Schrödinger equation Mathematical Physics Mathematical physics |
| Popis: | A useful semiclassical method to calculate eigenfunctions of the Schroedinger equation is the mapping to a well-known ordinary differential equation, as for example Airy's equation. In this paper we generalize the mapping procedure to the nonlinear Schroedinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schroedinger equation is mapped to the second Painleve equation, which is one of the best-known differential equations with a cubic nonlinearity. A quantization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential. |
| Databáze: | OpenAIRE |
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