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W. G. Pritchard Lab Seminar: 11:15 AM - 12:15 PM, 106 McAllister Bldg

	            **Monday February 6, 2006**
	
Stability, instability, and stability in deep-water surface waves

Diane Henderson

Department of Mathematics
Penn State University

Abstract:

Here we report experiments on permanent form gravity waves on deep water
propagating in both one and two horizontal dimensions. We find that
moderate amplitude, bi-periodic patterns are ``stable'' within the length
of our wave basin. This result is surprising in light of classic
instability results (the Benjamin-Feir instability) for deep-water waves.
And large amplitude experiments do show evidence of what appears to be the
Benjamin-Feir instability. However, recent numerical results by Fuhrman &
Madsen (2006) provide a different explanation. Our further experiments
show that their explanation is correct and the patterns are indeed
``stable''. To explain the unexpected persistence of these patterns
mathematically, we reconsider the stability of a uniform wavetrain using
the nonlinear Schroedinger (NLS) equation modified to include linear
damping.  We prove that the presence of damping, no matter how small,
stabilizes (with linear and nonlinear stability) the uniform wavetrain
solution. The predicted evolutions are in excellent agreement with our
experiments. These stability results are then extended to the case of a
permanent form solution of coupled NLS equations that model wave patterns.


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