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.

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