The modulation instability revisited |
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Authors: | H Segur D M Henderson |
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Institution: | (1) Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA;(2) Department of Mathematics, Penn State University, W.G. Pritchard Fluid Mechanics Laboratory, University Park, PA 16802, USA |
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Abstract: | The modulational instability (or “Benjamin-Feir
instability”) has been a fundamental principle of nonlinear wave propagation
in systems without dissipation ever since it was discovered in the 1960s. It
is often identified as a mechanism by which energy spreads from one dominant
Fourier mode to neighboring modes. In recent work, we have explored how
damping affects this instability, both mathematically and experimentally.
Mathematically, the modulational instability changes fundamentally in the
presence of damping: for waves of small or moderate amplitude, damping (of
the right kind) stabilizes the instability. Experimentally, we observe
wavetrains of small or moderate amplitude that are stable within the lengths
of our wavetanks, and we find that the damped theory predicts the evolution
of these wavetrains much more accurately than earlier theories. For waves of
larger amplitude, neither the standard (undamped) theory nor the damped
theory is accurate, because frequency downshifting affects the evolution in
ways that are still poorly understood. |
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