## 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  with AFM 

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.

## Constant Vorticity (Update)

In the quest to understand what happens to the stability as the strength of the vorticity increases, it’s important to predict precisely where the first instability will arise (in terms of the Floquet parameter $\mu$. Intersecting lines of opposite signature could give rise to a potential instability.  The two black dots follow the first to collisions; both of which yield instabilities for non-trivial solutions.

## AIMS 2018: Taipei (Day 4)

I’m exhausted. Today, I only attended one talk by Yuji Kodama on hydrodynamic hierarchies. I hadn’t really heard about these types of hierarchies before this talk. While I couldn’t really follow the entire talk, I did take away a few references.