五次Ginzburg-Landau方程的二维结构
Two-dimensional structures in the quintic Ginzburg-Landau equation
摘要：
利用Embry-Riddle航空大学的ZEUS集群，基于二维Fourier谱方法（Fourier空间离散化和时间差分显式格式）对具有三次和五次非线性项的二维复Ginzburg-Landau方程的时空孤子的存在范围进行了广泛的数值模拟.我们从Akhmediev { \it et. al.}并逐个缓慢地改变它们以确定孤子作为稳定/不稳定结构存在的区域。我们提出了八类耗散孤子，其中六个是已知的（固定的，脉动的，涡流旋转，灯丝，爆炸，蠕动）和两个是新的（蠕动涡流螺旋桨和旋转“豆形”孤子）。通过对方程的不同参数进行长时间的模拟，我们发现了稳定结构（静止、脉动、圆形涡旋旋转、有组织的爆炸）和不稳定结构（导致细丝的椭圆涡旋旋转、无组织的爆炸、蠕变）的存在范围。此外，通过改变甚至两个初始条件与涡量，我们发现一个更丰富的行为的形式蠕动涡螺旋桨，旋转的“豆形”孤子。每一个类区别于其他的独特功能，他们的能量演化，形状的初始条件，以及域的存在参数。
英文摘要：
By using ZEUS cluster at Embry-Riddle Aeronautical University we perform extensive numerical simulations based on a two-dimensional Fourier spectral method Fourier spatial discretization and an explicit scheme for time differencing) to find the range of existence of the spatiotemporal solitons of the two-dimensional complex Ginzburg-Landau equation with cubic and quintic nonlinearities. We start from the parameters used by Akhmediev {\it et. al.} and slowly vary them one by one to determine the regimes where solitons exist as stable/unstable structures. We present eight classes of dissipative solitons from which six are known (stationary, pulsating, vortex spinning, filament, exploding, creeping) and two are novel (creeping-vortex propellers and spinning "bean-shaped" solitons). By running lengthy simulations for the different parameters of the equation, we find ranges of existence of stable structures (stationary, pulsating, circular vortex spinning, organized exploding), and unstable structures (elliptic vortex spinning that leads to filament, disorganized exploding, creeping). Moreover, by varying even the two initial conditions together with vorticity, we find a richer behavior in the form of creeping-vortex propellers, and spinning "bean-shaped" solitons. Each class differentiates from the other by distinctive features of their energy evolution, shape of initial conditions, as well as domain of existence of parameters.
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