Cubic nonlinear Schrödinger equation with vorticity

In this paper, we introduce a new class of nonlinear Schrödinger equations (NLSEs), with an electromagnetic potential (A Φ), both depending on the wavefunction ψ. The scalar potential π depends on |ψ| 2, whereas the vector potential A satisfies the equation of magnetohydrodynamics with coefficient depending on ψ. In Madelung variables, the velocity field comes to be not irrotational in general and we prove that the vorticity induces dissipation, until the dynamical equilibrium is reached. The expression of the rate of dissipation is common to all NLSEs in the class. We show that they are a particular case of the one-particle dynamics out of dynamical equilibrium for a system of N identical interacting Bose particles, as recently described within stochastic quantization by Lagrangian variational principle. The cubic case is discussed in particular. Results of numerical experiments for rotational excitations of the ground state in a finite two-dimensional trap with harmonic potential are reported. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.