ALEX 2018 Workshop: Abstracts

From the Benjamin–Feir instability to Whitham modulation theory and beyond

Thomas J. Bridges

University of Surrey, Department of Mathematics (UK)

The talk is composed of three parts each of 10-15 minutes. In the first part I will talk about meeting Alexander and the work that emerged on the proof of the Benjamin-Feir (BF) instability [1] and its context. It was the first rigorous proof of the BF instability and continues to have impact in the theory of water waves.

In the second part of the talk two recent important applications of BF instability will be discussed. Firstly, the “Benjamin-Feir index” [2], which is a measure of the local strength of the instability, is being used to construct probability maps of the north Atlantic ocean for forecasting rogue waves, and is applied commercially to the routing of ships. Secondly, a remarkable theory of Elena Tobisch will be discussed whereby the BF instability is used to initiate an energy cascade in conservative systems [3]. It is a mechanism for generating a continuous energy spectrum and a highly complex wave field, starting with a BF instability. Both are having a big impact in applications but are in need of mathematical characterization and analysis.

In the third part of the talk I will discuss recent work which gives a new take on the BF instability. Nonlinear waves, such as Stokes periodic travelling water waves, can be modulated using Whitham theory. However, when characteristics coalesce this theory breaks down. A new theory shows that nonlinear modulation of this coalescence generates a new asymptotically valid two-way Boussinesq equation [4, 5]. The stabilization of the BF instability can be characterized as the implication of coalescing characteristics in Whitham theory. Connecting these two theories, it is then shown that the BF stabilization in shallow water, proved in [1], generates a nonlinear asymptotically-valid two-way Boussinesq equation. This latter discovery is in contrast to the two-way Boussinesq equation proposed by Boussinesq himself for shallow water waves which has been shown to be invalid!

Acknowledgments: The early work [1] was supported by a Fellowship from the Humboldt Foundation held at Stuttgart and Hannover, and the more recent work [4] was supported by the UK EPSRC under grant number EP/L505092/1.