The dynamics of interacting multi-pulses in the one-dimensional quintic complex Ginzburg-Landau equation

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.