Multidimensional self-trapping in linear and nonlinear potentials
Boris Malomed
备注：
to be published in volume "Models and Methods for Quantum Condensation and Fluids" (Lecture Notes Series of the Institute for Mathematical Sciences, National Univrsity of Singapore), editors W. Bao, Y. Cai, and I. Danalia
Solitons are typically stable objects in 1D models, but their straightforward extensions to 2D and 3D settings tend to be unstable. In particular, the ubiquitous nonlinear Schroedinger (NLS) equation with the cubic self-focusing, creates only unstable 2D and 3D solitons, because the same equation gives rise to the critical and supercritical collapse in the 2D and 3D cases, respectively. This article offers, first, a review of relevant settings which, nevertheless, make it possible to create stable 2D and 3D solitons, including ones with embedded vorticity. The main stabilization schemes considered here are: (i) competing (e.g., cubic-quintic) and saturable nonlinearities; (2) linear and nonlinear trapping potentials; (3) the Lee-Huang-Yang correction to the BEC dynamics, leading to the formation of robust quantum droplets; (4) spin-orbit-coupling (SOC) effects in binary BEC; (5) emulation of SOC in nonlinear optical waveguides, including PT-symmetric ones. Further, a detailed summary is presented for results which demonstrate the creation of stable 2D and 3D solitons by schemes based on the usual linear trapping potentials or effective nonlinear ones, which may be induced by means of spatial modulation of the local nonlinearity strength. The latter setting is especially promising, making it possible to use self-defocusing media, with the local nonlinearity strength growing fast enough from the center to periphery, for the creation of a great variety of stable multidimensional modes. In addition to fundamental ones and vortex rings, the 3D modes may be hopfions, i.e., twisted vortex rings with two independent topological charges. Many results for the multidimensional solitons are obtained, in such settings, not only in a numerical form, but also by means of analytical methods, such as the variational and Thomas-Fermi approximations.
Optical vortex trapping and annihilation by means of nonlinear Bessel beams in nonlinearly absorbing media
在非线性克尔介质中，当多光子吸收显著时，高阶非线性贝塞尔光束（NBB）中心的拓扑电荷m的涡旋可以稳定存在，并且无休止地存在。我们表明，m-充电NBB不仅是稳定的，但自发地形成从任何其他n-充电NBB和N“外国”旋涡的总电荷s随机嵌套在束截面，如果n + s = m。所有嵌套的旋涡合并在中心的原始NBB，这经历了模式转换的NBB，保持拓扑电荷和向内指向的功率电流，维持无衍射和无衰减的传播介质中的非线性吸收。我们预见到不同的应用，如创建稳定的，多电荷的旋涡没有严格的对齐要求，但通过自发的旋涡组合，混合波或粒子的旋涡可以引导，快速湮灭的旋涡偶极子，和清洁斑点光束的大量湮灭的旋涡。
