二维复五次金兹堡-朗道方程中脉冲的动态相互作用与平衡构型
Dynamic interactions and equilibrium configurations of pulses in the two-dimensional complex quintic Ginzburg-Landau equation
AI总结：
本文构建了一种快速有效的数值方案，精确计算二维复五次金兹堡-朗道方程中弱相互作用脉冲的动力学，通过全局中心流形约化得到快慢系统，并研究了N=2至5个脉冲的平衡解，包括稳定不动点和极限环。
摘要：
本文构建了一种快速有效的新型数值方案，精确计算二维复五次金兹堡-朗道方程（QCGLE）中弱相互作用脉冲的动力学。该数值方案使用全局中心流形约化，其中QCGLE的解被构造为单个脉冲之和加上一个余项函数，该余项函数被选为与QCGLE线性算子的零伴随本征模正交。将此构造解投影到稳定中心流形上，得到一个快慢系统，包含描述单个脉冲位置和相位的慢速常微分方程，以及描述余项函数的快速偏微分方程。通过考虑脉冲良好分离，系统可渐近展开为小参数$\epsilon=e^{-\lambda_r d}\ll1$，其中$\lambda_r$是脉冲的空间衰减率，$d>0$是最小脉冲分离距离。这里余项函数通过一个稳态偏微分方程确定，该方程可使用GMRES高效求解。考虑了$N=2,3,4$和$5$个脉冲的结果，发现存在不同的平衡解，如稳定不动点和极限环。还考虑了两个稳定的$N=3$相干结构的相互作用，发现常见趋势是结构退化为脉冲对，并从初始脉冲构型传播开去。
英文摘要：
This paper constructs a fast and effective novel numerical scheme which accurately calculates the dynamics of weakly-interacting pulses in the two-dimensional quintic-complex Ginzburg-Landau equation (QCGLE). The numerical scheme uses a global centre-manifold reduction, where the solution to the QCGLE is constructed as the sum of the individual pulses plus a remainder function, which is chosen to be orthogonal to the zero adjoint eigenmodes of the QCGLE linear operator. Projecting this constructed solution onto the stable centre-manifold leads to a fast-slow system of equations consisting of {\it slow} ordinary differential equations for the position and phases of the individual pulses and a {\it fast} partial differential equation for the remainder function. By considering the pulses to be well-separated, the system can be expanded asymptotically in terms of the small parameter $ε=e^{-λ_r d}\ll1$, where $λ_r$ is the spatial decay rate of the pulse, and $d>0$ is the minimum pulse separation distance. Here the remainder function is determined via a stationary partial differential equation that can be readily solved in an efficient manner using GMRES. Results for $N=2,3,4$ and $5$ pulses are considered, and it is found that different equilibrium solutions are possible such as stable fixed points and limit cycles. The interaction of two stable $N=3$ coherent structures is also considered, where the common tendency found is for the structure to degenerate into pairs of pulses which propagate away from the initial configuration of pulses.
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