Journal of Fluid Mechanics



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Stabilizing the Benjamin–Feir instability


HARVEY SEGUR a1, DIANE HENDERSON a2, JOHN CARTER a3, JOE HAMMACK a2 1 , CONG-MING LI a1, DANA PHEIFF a2 and KATHERINE SOCHA a4
a1 Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
a2 William G. Pritchard Fluid Mechanics Laboratory, Department of Mathematics, Penn State University, University Park, PA 16802, USA
a3 Department of Mathematics, Seattle University, Seattle, WA 98122-4340, USA
a4 Mathematics Department, Lyman Briggs School, Michigan State University, East Lansing, MI 48825-1107, USA

Article author query
segur h   [Google Scholar] 
henderson d   [Google Scholar] 
carter j   [Google Scholar] 
hammack j   [Google Scholar] 
li c   [Google Scholar] 
pheiff d   [Google Scholar] 
socha k   [Google Scholar] 
 

Abstract

The Benjamin–Feir instability is a modulational instability in which a uniform train of oscillatory waves of moderate amplitude loses energy to a small perturbation of other waves with nearly the same frequency and direction. The concept is well established in water waves, in plasmas and in optics. In each of these applications, the nonlinear Schrödinger equation is also well established as an approximate model based on the same assumptions as required for the derivation of the Benjamin–Feir theory: a narrow-banded spectrum of waves of moderate amplitude, propagating primarily in one direction in a dispersive medium with little or no dissipation. In this paper, we show that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation (of a certain type) stabilizes the instability. We arrive at this stability result first by proving it rigorously for a damped version of the nonlinear Schrödinger equation, and then by confirming our theoretical predictions with laboratory experiments on waves of moderate amplitude in deep water. The Benjamin–Feir instability is often cited as the first step in a nonlinear process that spreads energy from an initially narrow bandwidth to a broader bandwidth. In this process, sidebands grow exponentially until nonlinear interactions eventually bound their growth. In the presence of damping, this process might still occur, but our work identifies another possibility: damping can stop the growth of perturbations before nonlinear interactions become important. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.

(Published Online September 5 2005)
(Received February 19 2004)
(Revised April 6 2005)



Footnotes

1 Professor Hammack died during the publication process of this paper.



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