The Dynamics of Interacting Multi-pulses in the One-dimensional Quintic Complex Ginzburg-Landau Equation

Abstract

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 center-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 linearized operator about a single pulse. This center-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 ∊ = e-λrd0 where λr is a constant and d0 > 0 is the minimal 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 behavior, separated by an infinite set of heteroclinic orbits, are shown to "split" under perturbation creating complex spiral behavior. For the case of three-pulse interaction a range of dynamics, including chaotic pulse interaction, is 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.