Stabilizing the Benjamin–Feir instability

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.