Anomalous exponents and dipole solutions for the thin film equation

We investigate similarity solutions of the "thin film" equation. In particular we look at solutions on the half-line x ≥ 0 with compact support and zero contact angle boundary conditions in x = 0. Such "dipole" solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.