Dynamic interactions and equilibrium configurations of pulses in the two-dimensional complex quintic Ginzburg-Landau equation

TL;DR

This paper introduces a novel numerical scheme for efficiently simulating pulse interactions in the 2D complex quintic Ginzburg-Landau equation, revealing various equilibrium states and dynamic behaviors of pulse configurations.

Contribution

A new numerical method based on center-manifold reduction for accurately modeling weakly-interacting pulses in the 2D QCGLE, including analysis of equilibrium solutions and pulse dynamics.

Findings

Multiple equilibrium solutions including fixed points and limit cycles.

Interaction dynamics show pulses can form stable structures or propagate away.

Efficient computation of pulse interactions using the proposed scheme.

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 centre-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 centre-manifold leads to a fast-slow system of equations consisting of {\it slow} ordinary differential equations for the position and phases of the individual pulses and a {\it 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 $\epsilon=e^{-\lambda_r d}\ll1$, where $\lambda_r$ is the spatial decay rate of the pulse, and $d>0$ 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 $N=2,3,4$ 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 $N=3$ coherent structures is also considered, where the common tendency found is for the structure to degenerate into pairs of pulses which propagate away from the initial configuration of pulses.