Reverse draining of a magnetic soap film — Analysis and simulation of a thin film equation with non-uniform forcing

We analyze and classify equilibrium solutions of the one-dimensional thin film equation with no-flux boundary conditions and in the presence of a spatially dependent external forcing. We prove theorems that shed light on the nature of these equilibrium solutions, guarantee their validity, and describe how they depend on the properties of the external forcing. We then apply these results to the reverse draining of a one-dimensional magnetic soap film subject to an external non-uniform magnetic field. Numerical simulations illustrate the convergence of the solutions towards equilibrium configurations. We then present bifurcation diagrams for steady state solutions. We find that multiple stable equilibrium solutions exist for fixed parameters, and uncover a rich bifurcation structure to these solutions, demonstrating the complexity hidden in a relatively simple looking evolution equation. Finally, we provide a simulation describing how numerical solutions traverse the bifurcation diagram, as the amplitude of the forcing is slowly increased and then decreased.