Cubic Nonlinear Schrödinger Equation with vorticity

In this paper, we introduce a new class of nonlinear Schrödingerequations (NLSEs), with an electromagnetic potential (A, 8), both depending onthe wavefunction 9. The scalar potential 8 depends on |9|2, whereas the vectorpotential A satisfies the equation of magnetohydrodynamics with coefficientdepending on 9.In Madelung variables, the velocity field comes to be not irrotational ingeneral and we prove that the vorticity induces dissipation, until the dynamicalequilibrium is reached. The expression of the rate of dissipation is common toall NLSEs in the class.We show that they are a particular case of the one-particle dynamics out ofdynamical equilibrium for a system of N identical interacting Bose particles,as recently described within stochastic quantization by Lagrangian variationalprinciple.The cubic case is discussed in particular. Results of numerical experimentsfor rotational excitations of the ground state in a finite two-dimensional trap withharmonic potential are reported.