Wang C, Theocharis G, Kevrekidis P G, Whitaker N, Law K J H, Frantzeskakis D J, Malomed B A
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Oct;80(4 Pt 2):046611. doi: 10.1103/PhysRevE.80.046611. Epub 2009 Oct 30.
We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.
我们研究具有对称四阱势的二维非线性薛定谔/格罗斯 - 皮塔耶夫斯基（NLS/GP）方程中局域模的存在性和稳定性。利用相应的四模近似，我们从线性极限开始追踪捕获的定态模的参数演化，从而得出定态模族的完整分岔图。这给出了基本二维情形下自发对称性破缺的图景。在广泛的参数区域中，发现基于四模分解的预测与NLS/GP方程的全数值解吻合良好。在大振幅极限下，定态的稳定性性质与相应离散模型所暗示的性质一致。通过直接模拟探索不稳定模的动力学。最后，除了对自吸引非线性情形的全面分析外，还简要考虑了自排斥情形的分岔图。
