The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

The nonlinear Schrödinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrödinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross--Pitaevskii equation is discussed, which models the dynamics of Bose--Einstein condensates.