New models for multi-dimensional stable vortex solitons
New models for multi-dimensional stable vortex solitons
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DOI：
10.1007/s11467-018-0857-0
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发表时间：
2018-10
影响因子：
7.5
通讯作者：
H. Sakaguchi
中科院分区：
文献类型：
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作者：
H. Sakaguchi
孤子是稳定的孤波。1834年，斯科特·罗素首次在爱丁堡附近的一条狭窄的水道中观察到这种孤立的水波。1895年，D. Korteweg和G. de Vries导出了浅水表面波浪的Korteweg-de Vries （KdV）方程。1965年，N. Zabusky和M. Kruskal用数值模拟证明了Korteweg-de Vries方程中孤立波的稳定性，并创造了“孤子”一词。1967年，Gardner、Greene、Kruskal、Miura等人发现了逆散射变换的数学技术，系统地找到了KdV方程的解析解，KdV方程由此被认为是一个范式可积偏微分方程。非线性Schrödinger （NLS）方程是另一个非常重要的可积系统，其形式为i∂tϕ=(- 1/2)∂xxϕ+ g| ϕ| 2ϕ。该方程有平面波解，然而，三次自聚焦非线性项（g< 0）引起调制不稳定并产生孤波。NLS方程可以模拟深层无粘水表面的重力波和各种光学介质以及许多其他物理介质中的光波。许多关于孤子的论文和书籍已经出版[1-3]。最近，非线性Schrödinger方程被应用于超冷原子的玻色-爱因斯坦凝聚（BECs）。NLS方程中的三次非线性项来自于超冷原子之间的碰撞。在这种情况下，g对排斥和吸引相互作用分别为正和负。在7Li原子的玻色-爱因斯坦凝聚体中观察到具有吸引相互作用的孤子[4,5]。在玻色-爱因斯坦凝聚体的研究中，经常使用外部势来限制凝聚体。在这种情况下，具有外部电位的非线性Schrödinger方程称为Gross-Pitaevskii方程。一维NLS方程是一个可积系统，可以用逆散射法完全求解。然而，二维或三维NLS方程是不可积的。具有三次自聚焦非线性，g< 0的二维和三维NLS方程中存在一种奇异的坍缩现象[6,7]。在二维中，总范数有一个临界值N=
Solitons are stable solitary waves. In 1834, Scott Russel first observed such a solitary water wave in a narrow channel near Edinburgh. In 1895, D. Korteweg and G. de Vries had derived the Korteweg-de Vries (KdV) equations for waves on shallow water surfaces. In 1965, N. Zabusky and M. Kruskal had demonstrated the stability of the solitary waves in the Korteweg–de Vries equation using numerical simulations, and coined the term “soliton”. In 1967, Gardner, Greene, Kruskal, and Miura had discovered the mathematical technique of the inverse scattering transform to find analytical solutions to the KdV equation in a systematic manner, and the KdV equation was thus recognized as a paradigmatic integrable partial differential equation. The nonlinear Schrödinger (NLS) equation is another extremely important integrable system, which has a form of i∂ tϕ=(− 1/2)∂ xxϕ+ g| ϕ| 2ϕ. There are plane-wave solutions to this equation, however, the cubic self-focusing nonlinear term (with g< 0) causes modulational instability and creates solitary waves. The NLS equation can model gravity waves on the surface of deep inviscid water and light waves in various optical media, as well as many other physical media. A great many of papers and books about solitons have been published [1–3].More recently, the nonlinear Schrödinger equation was applied to the Bose–Einstein condensates (BECs) of ultracold atoms. The cubic nonlinear term in the NLS equation comes from collisions between the ultra-cold atoms. In this case, g is positive and negative for repulsive and attractive interactions, respectively. Solitons were observed experimentally in Bose-Einstein condensates of 7Li atoms with attractive interactions [4, 5]. In the studies of Bose-Einstein condensates, external potentials are often used to confine BECs. The nonlinear Schrödinger equation with external potentials are called the Gross–Pitaevskii equation in this context. The NLS equation in one dimension is an integrable system, which may be completely solved by the inverse-scattering method. However, the two-or three-dimensional NLS equation is not integrable. There is a singular phenomenon called collapse in two-and three-dimensional NLS equations with cubic self-focusing nonlinearity, g< 0 [6, 7]. In two dimensions, there is a critical value for the total norm, N=