Existence and large time behavior for a dissipative variant of the rotational NLS equation

We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates.We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.

Keywords

Gross-Pitaevskii equation, Bose-Einstein condensation, Ginzburg-Landau equation, stationary states, global attractor

2010 Mathematics Subject Classification

35B35, 35Q55, 35Q56, 37L15

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The first author is partly supported by Istituto Nazionale di Alta Matematica (GNAMPA Research Projects) and by the MUR Excellence Department Project awarded to GSSI, CUP D13C22003740001.

Received 15 September 2023

Received revised 9 January 2024

Accepted 10 January 2024

Published 18 July 2024