[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.

As for my viewpoint, all of my nonlocal operators are in a sense derived and come from reformulations of the problem of interest. For example, the Dirichlet-to-Neumann operator for free-surface water waves is an example of such a nonlocal operator.

This view point allows me to extend my nonlocal operator to a partial differential equation posed on some domain. Thus, by employing multiple scales before finding an implict form for the nonlocal operator in question, I can natually “bake them in from the start”.

Of course, there is a cost to doing this. For the DNO as described in the work of Ablowitz, Fokas, and Musslamini the implicit relationship for the DNO in the original variables is given by an integral over $$x$$. However, by converting to multiple scales before deriving the implicit relationship, the divergence form and resulting implicit relationship is replaced by an integral over each of the newly introduced scales $$\xi_0, \xi_1, \ldots, \xi_{n-1}$$ where I’ve been considering $\xi_j = \epsilon^j\left(x-c_j t\right), ~~j = 0, 1, \ldots, n-1, \qquad \tau = \epsilon^n t.$

This may not be the standard starting point for the various scales involved, but at the very least, it will allow me to consider NLS regimes more naturally.

I will admit, I may be approaching this from the wrong angle; but in all honesty, I found it difficult to parse the literature. When in doubt, I often return back to the basics to try and make sense of it on my own. At the very least, I have learned a bit in this process. For example, this seems to give a different (and perhaps easier method?) to identify 3- or 4-wave resonant interactions. This is an avenue I hope to pursue more in the near future.