Speaker: Harvey Segur
Title: The Modulational Instability with Weak Damping.
Affiliation: University of Colorado at Boulder.
Date: Friday, March 30, 2007.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. The modulational instability (sometimes called the Benjamin-Feir instability) has been a basic principle of nonlinear wave propagation ever since it was discovered in the 1960s. It can appear whenever waves propagate in a "dispersive" medium (i.e., a medium in which waves of different wavelength travel at different speeds) without dissipation. When the instability occurs, a uniform train of plane waves of finite amplitude is unstable. Examples of waves affected by this instability include surface waves on deep water, light waves in an optical fiber, Langmuir waves in a plasma, and waves in a Bose-Einstein condensate. (Dissipation is ignored in the mathematical model in each case). In terms of mathematical models, it applies whenever the equation governing the wave propagation is the well-known nonlinear Schroedinger equation, in the focussing case.

All of the physical examples listed above exhibit dissipation, although it is often weak. The non-intuitive result presented here is that even weak dissipation has a strong affect on the instability: Any amount of dissipation (of the right kind), no matter how small, stabilizes the instability. This conclusion is shown both mathematically and in physical experiments on waves on deep water. (The experiments were conducted by Diane Henderson and others at Penn State U.) A consequence is that some waves on deep water with two-dimensional, biperiodic surface patterns, described mathematically by Craig & Nicholls (2000, 2002) and experimentally by Hammack, Henderson & Segur (2005), are shown to be stable.