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 param...
| Main Authors: | , , , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2023
|
| _version_ | 1797111463499792384 |
|---|---|
| author | Chapman, SJ Dallaston, MC Kalliadasis, S Trinh, PH Witelski, TP |
| author_facet | Chapman, SJ Dallaston, MC Kalliadasis, S Trinh, PH Witelski, TP |
| author_sort | Chapman, SJ |
| collection | OXFORD |
| description | 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, <i>n</i>, 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 = ϵ1 > ϵ2 >⋯> 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. |
| first_indexed | 2024-03-07T08:10:46Z |
| format | Journal article |
| id | oxford-uuid:34571e6b-eb5d-4f03-b96e-b133aa5c3537 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:10:46Z |
| publishDate | 2023 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:34571e6b-eb5d-4f03-b96e-b133aa5c35372023-11-24T10:21:50ZThe role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamicsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:34571e6b-eb5d-4f03-b96e-b133aa5c3537EnglishSymplectic ElementsElsevier2023Chapman, SJDallaston, MCKalliadasis, STrinh, PHWitelski, TPWe 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, <i>n</i>, 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 = ϵ1 > ϵ2 >⋯> 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. |
| spellingShingle | Chapman, SJ Dallaston, MC Kalliadasis, S Trinh, PH Witelski, TP The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title | The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title_full | The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title_fullStr | The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title_full_unstemmed | The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title_short | The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics |
| title_sort | role of exponential asymptotics and complex singularities in self similarity transitions and branch merging of nonlinear dynamics |
| work_keys_str_mv | AT chapmansj theroleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT dallastonmc theroleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT kalliadasiss theroleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT trinhph theroleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT witelskitp theroleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT chapmansj roleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT dallastonmc roleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT kalliadasiss roleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT trinhph roleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics AT witelskitp roleofexponentialasymptoticsandcomplexsingularitiesinselfsimilaritytransitionsandbranchmergingofnonlineardynamics |