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.

## The Conservation Law Formulation

Ignoring total derivatives and the likes, we can write the nonlocal formulation as

$\displaystyle \frac{d}{dt}\int\varphi(x,\eta,t)\:dx = -\frac{1}{2}\int\left(\phi\frac{d}{dx}\varphi_x + \psi\frac{d}{dn}\varphi_x\right)\:dx.$

Should we choose $\varphi = (x+iz)^n$, then we find that the above reduces to

$\displaystyle \frac{d}{dt}\int(x + i\eta)^n\:dx = -\frac{1}{2}\int\left(n(n-1)(x+i\eta)^{n-2}(\phi + i\psi)(1+i\eta_x)\right)\:dx.$

It’s critical to note that the above form can be expressed as

$\displaystyle \frac{d}{dt}\int(x + i\eta)^n\:dx = -\frac{1}{2}\int\left(n(n-1)(x+i\eta)^{n-2}\omega\zeta\right)\:dx$

where $\omega = \phi + i \psi$ and $\zeta = 1 + i \eta_x$ following the notation of Olver, 1983.

It is straight forward to show that the right-hand side does indeed create a closed differential and thus, the integral along the surface can be transformed to an integral along the fixed boundaries.  This seems to imply that all of the functions $(x + i\eta)^n$ are indeed conserved densities in the sense of Olver.  Well this is where I made that sign mistake!  But, it is still worth following.

Of course, two major questions arise from this.  First, it’s clear that this is not a direct route to the conserved quantities of Olver.  For example, conserved densities that include terms such as $\phi(x,\eta,t)$ will not come from this form directly.  However, these can be found via a suggestion in made in the post Relating the conservation laws to the Bernoulli Equation.

That said, we can still find the conservation laws that include terms of the form $t\phi(x,\eta)$ and the likes.  For example, if we start with $\varphi = (x + i z)^2$, we find

$\displaystyle \frac{d}{dt}(x^2 - \eta^2 + 2ix\eta)\:dx = -\int\left(\phi - \eta\frac{d}{dx}\psi + i\left(x \frac{d}{dx}\psi + \eta_x\phi\right)\right)\:dx$

where we’ve used some integration-by-parts.  Noting that $\frac{d}{dx}\psi = -\eta_t$, two of the terms are total derivatives with respect to $t$.  The remaining terms can be dealt with by playing the game with the $\phi$ terms to rewrite them as

$\displaystyle -\frac{d}{dt}\left(t\int \phi(1 + i \eta_x)\:dx\right) + t\frac{d}{dt}\int\phi(1 + i\eta_x)\:dx$

Then, we only need to deal with the term $\frac{d}{dt}\phi$ term since we know  that $\phi\eta_x$ is a conserved density ($T_1$ in the notation of Olver or see Relating …).  Differentiating $\phi$ with respect to $t$, we find $\phi_t + \eta_t\phi_z$ which will ultimately yield a closed differential minus a contribution from gravity.  That is, we find

$\displaystyle t\int\left(\frac{1}{2}\left(\phi_z^2 - \phi_x^2\right) - \eta_x\phi_x\phi_z\right)\:dx - tg\int\eta\:dx$

The first integral yields a closed differential and can be replaced with integrals along the fixed boundary.  The second integral can be manipulated into a closed form by making a new contribution of the form $-\frac{d}{dt}\frac{1}{2}gt^2\eta$.

Combining all of this information together yields a conserved density of the form

$\displaystyle T = (x + i\eta)^2 +\frac{1}{2}\eta^2 - ix\eta + t\phi(1 + i\eta_x) + \frac{1}{2}gt^2\eta.$

Taking the real and imaginary parts of the above integral yields $T_6$ and $T_5$ from Olver’s hierarchy.

To find $T_3$, simply take $\varphi = x + iz$.  Finding $T_1$ requires a different perspective as discussed in Relating ….

Thus far, I have been able to recreate $T_1, T_3, T_4, T_5,$ and $T_6$.