The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

We study a prototypical example in nonlinear dynamics where transition to self-similarity in a singular limit is fundamentally changed as a parameter is varied. Here, we focus on the complicated dynamics that occur in a generalised unstable thin-film equation that yields finite-time rupture. A parameter, n, is introduced to model more general disjoining pressures. For the standard case of van der Waals intermolecular forces, n = 3, it was previously established that a countably infinite number of self-similar solutions exist leading to rupture. Each solution can be indexed by a parameter,.epsilon =epsilon(1). >epsilon(2) > center dot center dot center dot > 0, and the prediction of the discrete set of solutions requires examination of terms beyond-all-orders in.. However, recent numerical results have demonstrated the surprising complexity that exists for general values of n. In particular, the bifurcation structure of self-similar solutions now exhibits branch merging as n is varied. In this work, we shall present key ideas of how branch merging can be interpreted via exponential asymptotics. Crown Copyright (c) 2023 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).