Ring modes supported by concentrated cubic nonlinearity

Elad Shamriz, Boris A. Malomed

Research output: Contribution to journalArticlepeer-review


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 δ 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 band-gap states in the nonlinear system. Approximating the δ 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, multipeak 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.

Original languageEnglish
Article number052203
JournalPhysical Review E
Issue number5
StatePublished - 5 Nov 2018


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