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 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 $ε=e^{-λ_r d}\ll1$, where $λ_r$ 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.

Figures (16)

• Figure 1: Plot of the cell 1 phase space for (\ref{['eqn:rbar2']}) and (\ref{['eqn:gbar2']}) in the $(\overline{r}\cos \overline{g},\overline{r}\sin \overline{g})$-plane for the parameters in (a) (\ref{['eqn:parameters1']}) and (b) (\ref{['eqn:parameters2']}).
• Figure 2: Plot of (a,c) ${\rm Re}(V)$ and (b,d) ${\rm Im}(V)$ panels (b,d) for the parameter values (\ref{['eqn:parameters1']}) in panels (a,b) and (\ref{['eqn:parameters2']}) in panels (c,d). In each plot, the solid blue line represents the two-dimensional result while the red dashed line give the corresponding one-dimensional result.
• Figure 3: (a) $\langle\Phi,\psi_{r_1^x}\rangle$ (solid lines) and $\langle\Phi,\psi_{r_2^x}\rangle$ (dashed lines) and (b) $\langle\Phi,\psi_{g_1}\rangle$ (solid lines) and $\langle\Phi,\psi_{g_2}\rangle$ (dashed lines) as a function of $|r_1^x-r_2^x|$. The black result signify the full numerical results and the red results signify the asymptotic result. In each case the inner product is multiplied by $e^{\lambda_i d_{12}}$ to remove the exponential decay of the inner product with pulse separation distance.
• Figure 4: (a) Blow up of the $\overline{r}(0)=2.56$ phase plane trajectory with leading order $\widehat{w}(x,y,t)$ effects included. The red circle denotes the trajectory starting point and the green square denotes the trajectory end point after one period. Position of the trajectory end point $\overline{r}(T)$ as a function of (b) $\log_{10}({\rm Err})$ and (c) $M=L/\Delta x$ defining convergence.
• Figure 5: Heat map of the real-part of the solution (a,c) $u(x,y)$ and the remainder function (b,d) $w(x,y)=\epsilon\widehat{w}(x,y)$ for the trajectory starting with $(\overline{r},\overline{g})=(2.56,\pi/2)$ at (a,b) $\widehat{t}=917.06$ where $\overline{g}=\pi/2$ and $\dfrac{{\rm d}\overline{g}}{{\rm d}\widehat{t}}<0$ and (c,d) $\widehat{t}=T=1834.96$ where $\overline{g}=\pi/2$ and $\dfrac{{\rm d}\overline{g}}{{\rm d}\widehat{t}}>0$. (e) Plot of $r_1^x(\widehat{t})$ (black line) and $r_2^x(\widehat{t})$ (red line) for this trajectory showing the drift of the pulses to positive $x$ values over each period of the phase.