Ring modes supported by concentrated cubic nonlinearity

Abstract

We consider the one-dimensional Schrödinger equation on a ring, with the cubic term, of either self-attractive or repulsive sign, confined to a narrow segment. This setting can be realized in optics and Bose-Einstein condensates. For the nonlinearity coefficient represented by the delta-function, all stationary states are obtained in an exact analytical form. The states with positive chemical potentials are found in alternating bands for the cases of the self-repulsion and attraction, while solutions with negative chemical potentials exist only in the latter case. These results provide a possibility to obtain exact solutions for bandgap states in the nonlinear system. Approximating the delta-function by a narrow Gaussian, stability of the stationary modes is addressed through numerical computation of eigenvalues for small perturbations, and verified by simulations of the perturbed evolution. For positive chemical potentials, the stability is investigated in three lowest bands. In the case of the self-attraction, each band contains a stable subband, the transition to instability occurring with the increase of the total norm. As a result, multi-peak states may be stable in higher bands. In the case of the self-repulsion, a single-peak ground state is stable in the first band, while the two higher ones are populated by weakly unstable two-and four-peak excited states. In the case of the self-attraction and negative chemical potentials, single-peak modes feature instability which transforms them into persistently oscillating states.

Current production in ring condensates with a weak link

“…Current dynamics have been studied through either a rotational drive [21], through the interaction between symmetry breaking potentials and rotation such as in lattice rings [22][23][24], or through rotating defects [25][26][27]. Solutions of the Gross-Pitevskii equation (GPE) for a 1D ring, in the free case and with various sets of potentials, have been established by analyzing its spectrum either numerically and/or through the use of Jacobi elliptic functions [28][29][30][31][32][33][34]. The spectrum for a moving link and repulsive interactions was analyzed thoroughly in [35].…”

Oscillations and Decay of Superfluid Currents in a One-Dimensional Bose Gas on a Ring

“…Depending on the parameters, the barrier can resonantly couple the +1 and -1 states (light gray arrow) or induce an adiabatic transition between the +1 and 0 states (dashed blue arrow).force [27]. For sufficiently small obstacles stationary circulating states may exist [28][29][30], while a forced flow past a larger obstacle results in soliton emission [31][32][33].Most of the previous studies were performed in a rotating frame, thus imposing a flow onto the ring, allowing to estimate the nucleation rate of phase-slips [34]. For intermediate to strong interactions and small barriers it has been shown that the decay of persistent currents is related to the low-energy excitations in the ring [35].In this work, we investigate how a free current flows in 1D: as illustrated in Fig.…”

“…For a microscopic impurity the decay rate has been estimated by computing the drag force [27]. For sufficiently small obstacles stationary circulating states may exist [28][29][30], while a forced flow past a larger obstacle results in soliton emission [31][32][33].…”