Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping

Abstract

We consider the cubic nonlinear Schrödinger equation with harmonic trapping on R D (1 Ä D Ä 5). In the case when all directions but one are trapped (aka "cigar-shaped trap"), we prove modified scattering and construct modified wave operators for small initial and final data, respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on R D 1 . In the physical dimension D D 3, this system turns out to be exactly the (CR) equation derived by Faou, Germain, and the first author as the large box limit of the resonant NLS equation in the homogeneous (zero potential) setting. The special dynamics of the latter equation, combined with the above modified scattering results, allow us to justify and extend some physical approximations in the theory of Bose-Einstein condensates in cigar-shaped traps.

Asymptotic behavior of solutions to the cubic coupled Schrödinger systems in one space dimension

“…Using the method presented in [5], similar results have been obtained by adding a potential (see [4]), a harmonic trapping (see [6]), or by considering different derivatives along the Euclidean direction and the periodic one (see [12]). In the last mentioned article, Xu exhibits a scattering between the Schrödinger equation i∂ t U + ∆ R U − |∇ T |U = |U | 2 U in the spatial domain R × T and the cubic Szegő equation.…”

Multiplicity and stability of standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential

“…α i > 0 ( i = 1,2,3), which is called the anisotropy factors of the trap in quantum physics and trapping frequency of the i th‐direction in mathematics, see, eg, other works. ()…”

“…, which is called the anisotropy factors of the trap in quantum physics and trapping frequency of the ith-direction in mathematics, see, eg, other works. [20][21][22][23] Throughout this paper, we use standard notations. For simplicity, we write…”

Out-of-equilibrium dynamics and statistics of dispersive PDE

“…Finally, in a joint work with Laurent Thomann [28], we exhibit the dynamics of (CR) in a completely independent fashion, as an asymptotic system for NLS with partial harmonic trapping 4 :…”