Computing Multi-Localised Structures for Some Parabolic PDE Systems.

Localised structures such as fronts or pulses appear as equilibrium solutions in a variety of Partial Differential Equations (PDEs) that have been used to study a wide range of applied problems, from pulse propagation of impulses in nerve fibers in mathematical biology to pattern formation in nonlinear optics. This thesis is concerned with developing a numerical scheme for the computation of multiple co-existing localised structures (known as multi-localised structures). Studying multi-localised structures is numerically challenging since well-separated states interact via their exponentially decaying tails and numerical schemes struggle to capture these small interactions. This weak interaction leads to a slow movement of the states and so to study the movement one has to evolve the corresponding PDE for a large time. Standard time-stepping schemes are therefore expensive since one has to implement small tolerances to capture the correct dynamics. We develop an efficient and robust numerical scheme to compute multi-localised solutions in general PDEs that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts/pulses plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral Eigenmodes of each localised solution. Such a scheme efficiently captures the weakly interacting tails of the solutions allowing us to develop a fast time-stepping method. Furthermore, as the localised solutions become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then apply our numerical scheme to various real Ginzburg Landau equations where we observe a variety of behaviours, from colliding kinks to kinks converging to bound states. The comparison of our projection scheme (PS) with a standard time stepper scheme (SS) yields some interesting results that showcase the accuracy and robustness of PS. Moreover we provide a computational time comparison for several cases. We also apply the scheme to the quintic complex Ginzburg-Landau equation (QCGLE) in order to investigate the dynamics for several multi-pulse solutions. Using PS we are able, for first time, to capture dynamics for well-separated pulses and draw the full picture of the projected plane for two-pulse interaction in QCGLE. By using PS we observe limit circles on the QCGLE projected plane for two-pulse interaction. Furthermore, we study interaction between three pulses for different initial conditions. Finally, we discuss the importance of the results in the thesis and how the PS can be extended to general PDE systems and other multi-localised structures. Moreover, we go through some future work that can be done to improve the PS