Abstract

For many decades, the Benjamin-Feir instability of water waves has long attracted interest for its many dynamical intricacies. One such phenomenon is frequency downshifting, where the spectral peak of the wavetrain permanently decreases during the evolution of the instability. Long thought to be a viscously driven effect, a growing amount of evidence suggests that such phenomenon does not require dissipation. Is it possible to explain how permanent conservative downshifting is possible?  This talk revisits Whitham’s seminal work on the water wave problem from a phase dynamics perspective to answer this question. In doing so, we uncover that the role of the bulk mode is fundamental in the evolution of the Stokes waves. Further, we demonstrate that the Benjamin-Feir transition causes a point-wise linear degeneracy, which is the mechanism for frequency downshifting to be supported by the system. Such a conclusion is shown to be universal for Stokes waves with mean flow effects, providing an interesting outlook for the Benjamin-Feir instability in a menagerie of other systems, both hydrodynamic and beyond.  This talk is joint work with Tom Bridges (U. Surrey) and Olga Tritchenko (U. Western Ontario)