Localized travelling-wave solutions of partial differential equations on the real line correspond to homoclinic orbits of an associated ordinary differential equation. Here, we will focus on the numerical computation of homoclinic orbits and the accurate detection of their bifurcation points. Super-convergence results for the wave speed are presented. Finally, we will study the interaction of pulse packets. These are solutions of the PDE to an initial value which is given by concatenated, widely separated copies of a primary travelling wave. The fate of such solutions under truncation of the real line to a finite interval is investigated.

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