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 Olverâ€™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.

## The Dispersion Relationship for N-layers of Linear Shear

It would be interesting to use the AFM formulation to quickly derive not only the dispersion relationship for N-layers of linear shear, but also develop numerical simulations of traveling waves and relationships between the pressure and other internal wave properties.

## Constant Vorticity – The Saga Continues…

The above picture shows the real part of the growth rate $\lambda$ vs. the Floquet parameter $\mu$ for increasing truncation of the Floquet/Fourier/Hill matrix.

After a lengthy discussion with Richard Kollar, we have agreed that there seem to be two separate issues here.  The first is that one must be careful when restricting the Floquet Multiplier $\mu \in \left[0,\frac{1}{2}\right]$ in conjunction with the size of the truncated matrix for the Fourier/Floquet/Hill method.  The second is that there is still some other type of instability that may be present (the “moving” instability).

Continue reading “Constant Vorticity – The Saga Continues…”

## Thinking about the Gardner Transformation

In between cleaning, packing, and freaking out about my upcoming sabbatical, I’ve been thinking more about the conservation laws problem and specifically how they connect back to the KdV equation.

Warning: the following is completely unedited rambling!

## Possible New Recursion?

I’m still thinking about these conservation laws and finding a systematic method that doesn’t rely on identifying underlying symmetries in the problem.

More specifically, I’m trying to recreate the conservation laws from Olver without using the symmetry arguments directly.  After many sign errors, I can now create all of the conservation laws.  Unfortunately, it’s not a systematic process and relies on many clever insights.