2026-09-22T07:36:24Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/127490 2026-02-03 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2026-12-01 The student, Ting-Yang Hsiao, accepted the attached license on 2024-12-03 at 17:11. The student, Ting-Yang Hsiao, submitted this Dissertation for approval on 2024-12-03 at 17:20. This Dissertation was approved for publication on 2024-12-06 at 09:29. DSpace SAF Submission Ingestion Package generated from Vireo submission #21481 on 2025-03-28 at 14:55:37 Unstable stokes waves in constant vorticity flows Hsiao, Ting-Yang 2024-12-06 Hur, Vera Mikyoung Bronski, Jared Tzirakis, Nikolaos Zharnitsky, Vadim Water Waves Rotational Stokes Waves Modulational Instability eng We delve into the modulational instability of small-amplitude, non-zero vorticity Stokes waves of the classical water-wave problem, which concerns the two-dimensional incompressible flow of a perfect fluid of unit depth under the force of gravity. The stability analysis method is based on a periodic Evans function approach developed recently by Hur and Yang for irrotational Stokes waves [1, 2] and center manifold mechanism to reduce the infinite-dimensional hydrodynamic problem to a four-dimensional space. We construct a modulational instability index, denoted indlow(ω, κ), depending on the vorticity ω and the wave number κ, and prove instability when indlow(ω, κ) > 0. Our result reproduces the Benjamin–Feir instability for κ > 1.3627827 . . . when vorticity is zero. For the non-zero vorticity case, we depict the stability/instability region. This modulational instability result rigorously justifies the NLS approximation of Thomas, Kharif, and Manna [3] 2024-12 Thesis https://hdl.handle.net/2142/127490 Copyright 2024 Ting-Yang Hsiao Mathematics Mathematics University of Illinois at Urbana-Champaign Ph.D. Dissertation