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Unstable water waves: periodic Evans function approach

[Moved Online] Connections Workshop: Mathematical problems in fluid dynamics January 20, 2021 - January 22, 2021

January 20, 2021 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Vera Mikyoung Hur (University of Illinois at Urbana-Champaign)
Location: SLMath: Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Unstable Water Waves: Periodic Evans Function Approach
Abstract

I will discuss spectral instability of a Stokes wave of small amplitude in the finite depth. Analysis of a periodic Evans function, which is new, near the origin of the spectral plane offers an alternative proof of the Benjamin-Feir instability. Analysis near the resonance of order 2 reveals spectral instability when 0.86430...1.3627..., so new unstable waves are found. This seems the first rigorous proof of such high-frequency instability. I will discuss extensions to capillary-gravity waves and others, if time permits. Joint work with Z. Yang.
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Unstable Water Waves: Periodic Evans Function Approach

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