## Possible New Recursion?

I’m still thinking about these conservation laws and finding a systematic method that doesn’t rely on identifying underlying symmetries in the problem.

More specifically, I’m trying to recreate the conservation laws from Olver without using the symmetry arguments directly.  After many sign errors, I can now create all of the conservation laws.  Unfortunately, it’s not a systematic process and relies on many clever insights.

## Relating the conservation laws to the Bernoulli Equation

One of the interesting things that I realized when reading Olver, 1983 is his suggestion to investigate conservation laws as they relate to the quantities $\phi_n^m$ where

$\phi_n^m = \partial_x^n\partial_t^m\phi$

He also suggested looking at these quantities for the stream function $\psi$ as well.  Given that if $\phi$ is harmonic, so is $\phi_t$, I replaced $\phi$ with $\phi_t$ in the nonlocal equation.  This results is a more complicated expression given by (modulo a sign error or two)

$\displaystyle \frac{d}{dt}\int\phi\frac{d}{dn}\varphi_z\:dx = \int \phi_t \frac{d}{dx}\varphi_x - \eta_t \phi_x \frac{d}{dx}\varphi_z + \varphi_{zz}\eta_t^2\:dx$

Double check that there isn’t a typo in the above equation.

In this case, you are able to get the other conservation laws (specifically, ones that I haven’t been able to generate so far such as the impulse AKA $T_1$). This also gives a very straightforward way to incorporate the Bernoulli equation through the $\phi_t$ term on the right-hand side of the equation.

Of course, at this point, this isn’t taking into account any of the important simplification notes are used in Olver’s paper.  However, this is yet another way to consider attacking this problem.

## Connecting Olver [1983] with AFM [2006]

Thinking more about these conservation laws and the connection with the nonlocal formulation, consider a harmonic function $\varphi$ in conjunction with the velocity potential $\phi$ and the stream-function $\psi$.

It is straightforward to show that $(x + i\eta)^n$ are indeed conserved densities in the sense of Olver, 1983.  But do these make sense?  Well this was an embarrassing mistake, but the formulation is still helpful!  Details follow below.