## Congrats to Sal!

Last week, Salvatore Calatola-Young presented his Senior Synthesis project: “A Method for Deriving Conservation Laws for Water Waves”. In this talk, he presented the culmination of 3 years of research with me during his time as an undergraduate.

It wasn’t easy work, but he nailed it.

In summary, Sal showed how Olver’s conserved densities for the water-wave problem can be derived directly from a weak formulation without using the Lie Symmetries. Of course, Sal didn’t stop there! He extended the work for waves traveling over linear shear (constant vorticity), as well as multilayer flows. There’s a lot left to be explored here – and we’re working to bring it all together.

You can find the first in a series of papers available on the ArXiV (https://arxiv.org/abs/2105.07580).

## Finding Conservation Laws

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.

## Thinking more about multiple scales

[mathjax] Perhaps this is widely known, but I got a bit stuck-in-the-weeds thinking about applying multiple scales to problems with nonlocal operators. I have honestly been a bit unsatisfied with the literature that I have found on this scenario – that said, if you have any suggestions or references, please feel free to leave them in the comments! Continue reading to see what I’ve been thinking about.

## 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*}