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https://hdl.handle.net/2142/127490
Description
Title
Unstable stokes waves in constant vorticity flows
Author(s)
Hsiao, Ting-Yang
Issue Date
2024-12-06
Director of Research (if dissertation) or Advisor (if thesis)
Hur, Vera Mikyoung
Doctoral Committee Chair(s)
Bronski, Jared
Committee Member(s)
Tzirakis, Nikolaos
Zharnitsky, Vadim
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Water waves, Rotational Stokes waves, Modulational instability
Abstract
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]
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