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?

As I have mentioned in previous posts, I’ve always been fascinated by conservations laws. Interestingly, in their original paper, Ablowitz, Fokas, and Musslamini were able to generate a two conservation laws from their nonlocal equation by asymptotically expanding in \(k\). I was curious if I could find all of the conservation laws derived by Olver, 1983. And it turns out, you can!

Likewise, you can extend Olver’s results in a systematic manner for many other fluid configurations. Currently, I am working with Salvatore Calatola-Young on completing a series of papers to this end. We do this by recasting the dynamic boundary condition into a nonlocal form.

This short note shows how to recast the equations into this coupled system of nonlocal equations. Furthermore, it highlights a few other things:

  • Traveling Waves. When seeking traveling wave solutions, the equations collapse into a single equation. While there are many different “single traveling wave equations” for one-dimensional surfaces, this particular form has the advantage of being derivative free for waves without surface tension. Interestingly, finding the asymptotic approximation for periodic traveling-wave solutions by hand also seems easier due to the lack of derivative.
  • Time-Dependent Pressure. Using the nonlocal/nonlocal formulation, it is easy to get maps between boundary data. Specifically, once can easily write down a direct map from the pressure at the bottom to the time-dependent free-surface variables.
  • Easily Generating Different Asymptotic Formulae. In this slightly generalized framework, it is possible to create different asymptotic formula all valid to the same order. Depending on the use, one formulation my have significant advantages over the other.

In the coming week, I hope that Sal and I will have our paper posted with the connection to deriving all of Olver’s conservation laws, as well as other, suprising connections.

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