多维孤子：已确立的结果和新发现
Multidimensional solitons: Well-established results and novel findings
摘要：
简要回顾了一些著名的和一些最近的结果，在研究二维和三维（2D和3D）孤子。零涡（基本）孤子和携带涡S = 1的。并简要讨论了原子玻色-爱因斯坦凝聚体和非线性光学中的多维孤子的物理实现。与通常稳定的1D孤子不同，2D和3D孤子容易受到临界和超临界崩溃的发生所引起的不稳定性的影响，分别在引起孤子的相同的2D和3D模型中。涡旋孤子具有更强的分裂不稳定性。为此，一个中心问题是寻找可以稳定2D和3D孤子的物理设置。简短的审查涉及一个既定的主题，即，本文的主要工作包括：在谐振子势（HO势）中S = 0和1的三维和二维态的稳定性，以及最近发展起来的另一个课题：在自旋轨道耦合（SOC）的二元系统中，S = 0和（+/-）1的分量（半涡旋和混合模）并置的二维和三维自由空间孤子的稳定性。前者是基于单立方非线性薛定谔/Gross-Pitaevskii方程（NLSE/GPE），而后者是由两个耦合GPE的系统表示。在这两种情况下，2D和3D几何形状中的通用情况截然不同。在2D设置中，稳定化机制产生稳定的基态（GS，在没有稳定化的情况下不存在），其范数低于临界崩溃开始的阈值。在三维几何中，超临界坍缩不允许产生GS，但可以构造亚稳态孤子。
英文摘要：
A brief review is given of some well-known and some very recent results obtained in studies of two- and three-dimensional (2D and 3D) solitons. Both zero-vorticity (fundamental) solitons and ones carrying vorticity S = 1 are considered. Physical realizations of multidimensional solitons in atomic Bose-Einstein condensates (BECs) and nonlinear optics are briefly discussed too. Unlike 1D solitons, which are typically stable, 2D and 3D ones are vulnerable to instabilities induced by the occurrence of the critical and supercritical collapse, respectively, in the same 2D and 3D models that give rise to the solitons. Vortex solitons are subject to a still stronger splitting instability. For this reason, a central problem is search for physical settings in which 2D and 3D solitons may be stabilized. The brief review addresses one well-established topic, viz., the stabilization of the 3D and 2D states, with S = 0 and 1, trapped in harmonic-oscillator (HO) potentials, and another topic which was developed very recently: the stabilization of 2D and 3D free-space solitons, which juxtapose components with S = 0 and (+/-)1 (semi-vortices and mixed modes), in a binary system with the spin-orbit coupling (SOC) between its components. The former model is based on the single cubic nonlinear Schroedinger/Gross-Pitaevskii equation (NLSE/GPE), while the latter one is represented by a system of two coupled GPEs. In both cases, generic situations are drastically different in the 2D and 3D geometries. In the 2D settings, the stabilization mechanism creates a stable ground state (GS, which was absent without the stabilization), whose norm falls below the threshold value at which the critical collapse sets in. In the 3D geometry, the supercritical collapse does not allow to create a GS, but metastable solitons can be constructed.
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