We formulate an effective numerical scheme that can readily, and accurately, calculate the dynamics of weakly interacting multi-pulse solutions of the quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The scheme is based on a global centre-manifold reduction where one considers the solution of the QCGLE as the composition of individual pulses plus a remainder function, which is orthogonal to the adjoint eigenfunctions of the linearised operator about a single pulse. This centre-manifold projection overcomes the difficulties of other, more orthodox, numerical schemes, by yielding a fast-slow system describing 'slow' ordinary differential equations for the locations and phases of the individual pulses, and a 'fast' partial differential equation for the remainder function. With small parameter $\epsilon=e^{-\lambda_r d}$ where $\lambda_r$ is a constant and $d>0$ is the pulse separation distance, we write the fast-slow system in terms of first-order and second-order correction terms only, a formulation which is solved more efficiently than the full system. This fast-slow system is integrated numerically using adaptive time-stepping. Results are presented here for two- and three-pulse interactions. For the two-pulse problem, cells of periodic behaviour, separated by an infinite set of heteroclinic orbits, are shown to 'split' under perturbation creating complex spiral behaviour. For the case of three pulse interaction a range of dynamics, including chaotic pulse interaction, are found. While results are presented for pulse interaction in the QCGLE, the numerical scheme can also be applied to a wider class of parabolic PDEs.

翻译：我们提出一种有效的数值方案，能够简便且精确地计算一维五次复金兹堡-朗道方程（QCGLE）中弱相互作用的多脉冲解的动力学。该方案基于全局中心流形约化，将QCGLE的解视为单个脉冲的组合加上一个余函数，该余函数与线性化算子（围绕单个脉冲）的伴随特征函数正交。这种中心流形投影克服了其他更传统数值方案中的困难，通过生成一个快慢系统：描述单个脉冲位置和相位的“慢”常微分方程，以及描述余函数的“快”偏微分方程。鉴于小参数ε = e^{-λ_r d}（其中λ_r为常数，d>0为脉冲间距），我们将快慢系统仅用一阶和二阶修正项表示，这种形式的求解效率高于完整系统。该快慢系统采用自适应时间步进进行数值积分。本文给出了两脉冲和三脉冲相互作用的结果。对于两脉冲问题，由无限族异宿轨道分隔的周期性行为单元，在扰动下发生“分裂”，产生复杂的螺旋动力学。对于三脉冲相互作用，发现包括混沌脉冲相互作用在内的一系列动力学行为。虽然结果基于QCGLE中的脉冲相互作用，但该数值方案也可应用于更广泛的抛物型偏微分方程。