The computation and semi-analytic theory of standing deep water waves

We implement a modified version of the algorithm created by Amick and Toland to numerically compute the coefficients of the power series solutions to the classical standing water wave problem. With this algorithm, we precisely compute the coefficients to the 39th order of the wave amplitude. We also verify that Amick and Toland's algorithm is equivalent to Schwartz and Whitney's. We are unable to prove that the resulting power series solutions converge, although we propose a proof technique involving rigorous numerics and classical induction that may work for simpler problems. In lieu of a convergence proof, we estimate the solutions' radius of convergence, in the event that they do converge, using both truncated Taylor series and Pad\'e approximants. Additionally, we conjecture about the solutions' local behavior and analytic structure.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Ammons, Savana Jade

Contributors dc:contributor

• Hur, Vera Mikyoung
• Laugesen, Richard
• Bronski, Jared
• Tzirakis, Nikolaos

• Partial Differential Equations
• Differential Equations
• Water Waves
• Standing Waves
• Algorithms
• Numerical Analysis

Statement dc:rights

• Copyright 2025 Savana Ammons

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129534

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129534