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.

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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

\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, I take a look at extending their formulation to a coupled system of integro-differential equations given by

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?

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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.

2019 PNW SIAM Conferece – Hosted at Seattle U

This past weekend, Seattle University hosted the 2nd Biennial Meeting of SIAM Pacific Northwest Section. I’m happy to have been part of the organizing committee. In the end, we hosted 4 plenary speakers and 22 different sessions with over 140 participants.

It was a lot of work organizing this conference, but it could not have been done without the incredible support of Grady Wright (Boise State University) and John Carter (Seattle University).

Check out our At a Glance Schedule Here!