Ring modes supported by concentrated cubic nonlinearity

TL;DR

This paper derives exact analytical solutions for localized modes in a one-dimensional nonlinear Schrödinger equation on a ring with a concentrated cubic nonlinearity, analyzing their stability in optical and BEC contexts.

Contribution

It provides exact solutions for bandgap states with delta-function nonlinearity and investigates their stability through numerical and simulation methods.

Findings

Exact stationary states are obtained analytically for delta-function nonlinearity.

Stable multi-peak states exist in higher bands for self-attraction.

Ground state stability depends on the sign of the nonlinearity and chemical potential.

Abstract

We consider the one-dimensional Schroedinger 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.