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 $x\ge0$ 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.

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