标题： 集中立方非线性支持的环形模式 显示英文标题

标题： Ring modes supported by concentrated cubic nonlinearity

摘要： 我们考虑在环上的一维薛定谔方程，带有自吸引或排斥符号的三次项，并限制在一个狭窄的区间内。 这种设置可以在光学和玻色-爱因斯坦凝聚体中实现。 对于由δ函数表示的非线性系数，所有静态状态都可以以精确的解析形式获得。 对于自排斥和吸引的情况，正化学势的状态在交替带中被找到，而负化学势的状态仅在后一种情况下存在。 这些结果为在非线性系统中获得带隙态的精确解提供了可能性。 通过将δ函数近似为窄高斯函数，通过数值计算小扰动的特征值来研究静态模式的稳定性，并通过扰动演化的模拟进行验证。 对于正化学势，在三个最低带中进行了稳定性研究。 在自吸引的情况下，每个带包含一个稳定子带，随着总规范的增加，会发生向不稳定的转变。 因此，多峰态可能在较高带中稳定。 在自排斥的情况下，第一带中单峰基态是稳定的，而更高带中则填充了弱不稳定性的两峰和四峰激发态。 在自吸引和负化学势的情况下，单峰模式表现出不稳定性，这使它们转变为持续振荡的状态。 显示英文摘要

摘要： 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.