Carles, Rémi Critical nonlinear Schrödinger equations with and without harmonic potential. (English) Zbl 1029.35208 Math. Models Methods Appl. Sci. 12, No. 10, 1513-1523 (2002). Summary: Bose-Einstein condensation is usually modeled by a nonlinear Schrödinger equation with harmonic potential \[ i\hbar \partial_t \psi^\hbar+ {\hbar^2\over 2m}\Delta \psi^\hbar= {m\over 2}\omega^2 x^2\psi^\hbar +{4\pi \hbar^2a\over m}|\psi^\hbar |^2\psi^\hbar, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^n, \] where \(\omega>0\) and \(a\) is the scattering length, whose sign differs according to the chemical element considered.We use a change of variables that turns the critical nonlinear Schrödinger equation into the critical nonlinear Schrödinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on \(L^2\), and bijective on some time intervals. Using the known results for the critical nonlinear Schrödinger equation, this provides information for the properties of Bose-Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon. Cited in 51 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B33 Critical exponents in context of PDEs 35B35 Stability in context of PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 82D99 Applications of statistical mechanics to specific types of physical systems Keywords:critical nonlinear Schrödinger equation; harmonic potential; Bose-Einstein condensator; blow-up in finite time; wave collapse × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] DOI: 10.1103/PhysRevLett.78.985 · doi:10.1103/PhysRevLett.78.985 [2] DOI: 10.1103/PhysRevLett.75.1687 · doi:10.1103/PhysRevLett.75.1687 [3] DOI: 10.1512/iumj.2000.49.1804 · Zbl 0970.35143 · doi:10.1512/iumj.2000.49.1804 [4] DOI: 10.1007/BFb0086749 · doi:10.1007/BFb0086749 [5] DOI: 10.1103/PhysRevLett.85.1146 · doi:10.1103/PhysRevLett.85.1146 [6] DOI: 10.1007/BF00251502 · Zbl 0676.35032 · doi:10.1007/BF00251502 [7] DOI: 10.1215/S0012-7094-93-06919-0 · Zbl 0808.35141 · doi:10.1215/S0012-7094-93-06919-0 [8] DOI: 10.1007/BF02096981 · Zbl 0707.35021 · doi:10.1007/BF02096981 [9] DOI: 10.1002/cpa.3160450204 · Zbl 0767.35084 · doi:10.1002/cpa.3160450204 [10] Niederer U., Helv. Phys. Acta 47 pp 167– (1974) [11] DOI: 10.1016/0022-0396(89)90123-X · Zbl 0703.35158 · doi:10.1016/0022-0396(89)90123-X [12] DOI: 10.1103/PhysRevE.62.6224 · doi:10.1103/PhysRevE.62.6224 [13] DOI: 10.1007/BF01626517 · Zbl 0356.35028 · doi:10.1007/BF01626517 [14] DOI: 10.1007/BF01208265 · Zbl 0527.35023 · doi:10.1007/BF01208265 [15] DOI: 10.1023/A:1026437923987 · Zbl 0989.82024 · doi:10.1023/A:1026437923987 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.