En Route to Conservation Laws

Deriving conservation laws from the nonlocal formulation of Ablowitz, Fokas, and Musslimani has been a little bit of a pet project of mine for exactly 2 years now. Finally, things have come together and I’ve started documenting things. What I was hoping would be a short paper has now turned into a series of papers on various different topics. I just posted a short note ot the ArXiV that lays out the first step – recasting the AFM formulation into a coupled system of nonlocal equations that encorporate both the kinematic and dynamic boundary conditions. I’m hoping to have the remaining papers posted in the near future. Once the papers are approved on the ArXiV, I’ll post them here along with some exposition. Excited to finally have a lot of this documented.

Updates – in brief

It has been a while.  I’ve been working on quite a few projects with a few students.  I’ve made significant progress on the conservation laws project as well as the time-dependent pressure problem. 

Conservation Laws Project

I recently gave a talk on this in the “Waves in One World” Series.  You can find a direct link to the talk as well as the corrected  slides below.  I’ll write more information about this in an upcoming post once we have completed the pre-prints.  

Talk Abstract: We consider a nonlocal formulation of the water-wave problem for a free surface with an irrotational flow, and show how the problem can be reduced to a singleequation for the interface. The formulation is also extended to constant vorticity and interfacial flows of different density fluids. We show how this formulationcan be used to systematically derive Olver’s conservation laws not only for an irrotational fluid, but for constant vorticity and interfaces. This framework easily lends itself to computing the related conservation laws for various asymptotic models.

Waves in One World – Online Seminar – April 22, 2020
Recording of Talk
Annotated Slides (.pdf)

Time-Dependent Pressure Problem

Along the way with the conservation law problem, we discovered a direct map between the pressure at the bottom and the free-surface variables. I’ll detail more about this soon.