Self-Similarity in a Coarsening Model in One Dimension

Motivated from asymptotic laws of motion for transition layers in the equation u_t = ɛ ²u_xx + u-u³, we consider the following model for coarsening of a fine partition of an interval: find the shortest subinterval of the partition, and joint it with its neighbours, combining three into one. Making a `random order assumption', we develop and study an unusual coagulation equation for the distribution of interval lengths. We establish the existence of a self-similar solution of this equation by using Laplace transform techniques. Simulation data indicate that random order does persist if present initially, and the distribution approaches similarity form.