dorsal/arxiv
View SchemaUniform semiclassical approximations of the nonlinear Schroedinger equation by a Painleve mapping
| Authors | D. Witthaut, H. J. Korsch |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0608099 |
| URL | https://arxiv.org/abs/quant-ph/0608099 |
| DOI | 10.1088/0305-4470/39/47/012 |
| Journal | J. Phys. A: Math. Gen. 39, 14687 (2006) |
Abstract
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.
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"abstract": "A useful semiclassical method to calculate eigenfunctions of the Schroedinger\nequation is the mapping to a well-known ordinary differential equation, as for\nexample Airy\u0027s equation. In this paper we generalize the mapping procedure to\nthe nonlinear Schroedinger equation or Gross-Pitaevskii equation describing the\nmacroscopic wave function of a Bose-Einstein condensate. The nonlinear\nSchroedinger equation is mapped to the second Painleve equation, which is one\nof the best-known differential equations with a cubic nonlinearity. A\nquantization condition is derived from the connection formulae of these\nfunctions. Comparison with numerically exact results for a harmonic trap\ndemonstrates the benefit of the mapping method. Finally we discuss the\ninfluence of a shallow periodic potential on bright soliton solutions by a\nmapping to a constant potential.",
"arxiv_id": "quant-ph/0608099",
"authors": [
"D. Witthaut",
"H. J. Korsch"
],
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"quant-ph"
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"doi": "10.1088/0305-4470/39/47/012",
"journal_ref": "J. Phys. A: Math. Gen. 39, 14687 (2006)",
"title": "Uniform semiclassical approximations of the nonlinear Schroedinger equation by a Painleve mapping",
"url": "https://arxiv.org/abs/quant-ph/0608099"
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