Title: Modulational Instability of Viscous Fluid Conduit Periodic Waves.
Abstract: The Whitham modulation equations are widely used to describe the behavior of modulated periodic waves on large space and time scales; hence, they are expected to give insight into the stability of spatially periodic structures. However, the derivation of these equations are based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this talk, I aim to rigorously verify the predictions of the Whitham modulation equations in the context of the so-called conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, using rigorous spectral perturbation theory, the predictions of the Whitham modulation equations will be connected to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. This makes rigorous recent formal results on the conduit equation obtained by Maiden and Hoefer.