Eur. Phys. J. Special Topics 147, 25-43 (2007)
https://doi.org/10.1140/epjst/e2007-00201-1

The modulation instability revisited

¹ Department of Applied Mathematics, University of Colorado, Boulder, CO, 80309-0526, USA
² W.G. Pritchard Fluid Mechanics Laboratory, Department of Mathematics, Penn State University, University Park, PA, 16802, USA

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.