Dynamic Interactions and Equilibrium Configurations of Pulses in the Two-Dimensional Complex Quintic Ginzburg–Landau Equation

Abstract.

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 center-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 center-manifold leads to a fast-slow system of equations consisting of slow ordinary differential equations for the position and phases of the individual pulses and a 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 [Formula: see text], where [Formula: see text] is the spatial decay rate of the pulse, and [Formula: see text] 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 [Formula: see text], 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 [Formula: see text] coherent structures is also considered, where the common tendency is for the structure to degenerate into pairs of pulses which propagate away from the initial configuration of pulses.