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 quintic complex Ginzburg–Landau equation (QCGLE) in one space dimension. The is based on a global center-manifold reduction where considers solution QCGLE as composition individual pulses plus remainder function, which orthogonal to adjoint eigenfunctions linearized operator about single pulse. This projection overcomes difficulties other, more orthodox, schemes, by yielding fast-slow system describing “slow” ordinary differential equations for locations phases pulses, “fast” partial function. With small parameter constant minimal pulse separation distance, we write terms first-order second-order correction only, formulation solved efficiently than full system. integrated numerically using adaptive time-stepping. Results are presented here two- three-pulse interactions. For two-pulse problem, cells periodic behavior, separated infinite set heteroclinic orbits, shown “split” under perturbation creating spiral behavior. case interaction range dynamics, including chaotic interaction, found. While results QCGLE, also be applied wider class parabolic PDEs.