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

S Jonathan Chapman, Michael Dallaston, Serafim Kalliadasis, Philippe Trinh, Thomas Witelski

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2 Citations (SciVal)


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, ϵ=ϵ 12>⋯>0, and the prediction of the discrete set of solutions requires examination of terms beyond-all-orders in ϵ. 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.

Original languageEnglish
Article number133802
Number of pages31
JournalPhysica D : Nonlinear Phenomena
Early online date21 Jun 2023
Publication statusPublished - 30 Nov 2023

Bibliographical note

Funding Information:
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Applicable Resurgent Asymptotics when work on this paper was undertaken. This work was supported by EPSRC, United Kingdom Grant No. EP/R014604/1 . PHT gratefully acknowledges support from EPSRC, United Kingdom Grant No. EP/V012479/1 and also Lincoln College (Oxford), United Kingdom and the support of the Zilkha Fund . SK acknowledges support from EPSRC, United Kingdom Grants No. EP/K008595/1 and No. EP/L020564/1 , and ERC, via Advanced Grant No. 247031 .


  • Complex analysis
  • Exponential asymptotics
  • Fluid dynamics
  • Similarity solutions
  • Thin-film flows

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics


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