In joint work with Sal, we have finally written down the systematic process for deriving all of Olver’s conservation laws for the water-wave problem from our nonlocal-nonlocal formulation. This post outlines the current state of deriving the conservation laws from this **Nonlocal-Nonlocal** formulation discussed in the A Weak Formulation of Water Waves in Surface Variables. Click **Continue Reading…** to learn a little bit more about the process.

## A Weak Formulation of Water Waves in Surface Variables

In 2006, Ablowitz, Fokas, and Musslamini published a paper where they reformulated the water-wave problem for irrotational fluids as a system of equations in surface variables given by

[mathjax]\begin{align*}

\displaystyle 0&= q_t + \frac{1}{2}q_x^2 + g\eta -\frac{1}{2}\frac{(\eta_t+q_x\eta_x)^2}{1 + \eta_x^2}\\

\displaystyle 0&= \int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

In this formulation, \(q(x,t)\) is the trace of the velocity potential along the free surface \(z = \eta(x,t)\).

In A Weak Formulation of Water Waves in Surface Variables (short note posted on arXiv.org), I take a look at extending their formulation to a coupled system of integro-differential equations given by

\begin{align*}

0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\left(q_x\eta_t – \eta_x(q_t+g\eta)\right)\cosh(k(\eta+h)) -i\left(q_t + g\eta\right)\sinh(k(\eta+h))\right) dx\\

0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

You might be asking, *why? *

## 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.

Waves in One World – Online Seminar – April 22, 2020

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.

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.

## 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!*

Continue reading “Thinking about the Gardner Transformation”