For the past week, I’ve been really preoccupied with reviewing a paper and working with students on creating a conservation law hierarchy.

I’ve been surprised in how long it’s taking me to complete reviewing this paper.  In fact, I submitted a 2 page review, only to realize that I was incorrect.  Luckily, the editor hadn’t sent it to the reviewer yet.  I’m frantically trying to rewrite my review at this point.   I don’t think I’ve spent this long on a review.  At some point, I need to just call it.

As for the conservation law problem, one of my research student is making great progress.  In terms of thinking about the conservation law property, recall that for any harmonic function, we can write the nonlocal form of the water wave problem as

$\displaystyle \frac{d}{dt}\int_S \psi(x,\eta,t)\:dx = -\int_S\phi(x,\eta,t)\frac{d}{dx}\psi_x(x,\eta,t)\:dx + J_\Gamma.$

Here, I’m following the notation in Benjamin & Olver’s 1982 paper (more or less).  What’s important here is that we consider a hierarchy of harmonic functions $\Psi$ that should allow us to generate a list of conservation laws.

Using some sort of lexicographical ordering so that $\Psi^{(m)}<\Psi^{(n)}$ provided that $m (optimal ordering TBD), we can write the above integral relationship as

$\displaystyle \frac{d}{dt}\int_S \Psi^{(n)}(x,\eta,t)\:dx = -\int_S\phi(x,\eta,t)\frac{d}{dx}\Psi^{(n)}_x(x,\eta,t)\:dx + J_\Gamma.$

We really don’t care about the $J_\Gamma$ part for now.  These are bulk quantities.  Should we consider waves that decay at $\pm \infty$, these equations vanish.  I believe that Benjamin & Olver refer to these as ticklish details.  Looking at the list of conservation laws on pg 157 of the paper along with the definition (5.2), expressing the above as a conservation law requires “dealing with” the term

$\int_S\phi(x,\eta)\frac{d}{dx}\Psi^{(n)}_x(x,\eta)\:dx.$

Choosing $\Psi^{(1a)} = z$ quickly yields conservation of mass in the form

$\displaystyle \frac{d}{dt}\int_S\eta\:dx = J_\Gamma.$

This is precisely the form that we need in accordance with (5.2).  However, choosing $latex \Psi^{(2b)} = x z$ yields

$\displaystyle \frac{d}{dt}\int_S x\eta\:dx = -\int_S\phi(x,\eta)\eta_x\:dx + J_\Gamma.$

Playing a little game, the above can be rewritten as

$\displaystyle \int_S \frac{d}{dt}\left(x\eta + t\eta_x\phi(x,\eta,t)\right)- t\frac{d}{dt}\left(\phi(x,\eta)\eta_x\right) \:dx = J_\Gamma.$

The problematic term on the right-hand side is now replaced by the term

$t \int_S \frac{d}{dt}\left(\phi(x,\eta)\eta_x\right)\:dx = 0$

We know that this is zero since this is a separate conservation law that is a consequence of the Bernoulli Equation (referred to as the horizontal impulse).

So, we now have a new conserved density in given by $x\eta + t \eta_x\phi(x,\eta)$.    We can proceed in a similar manner generating other conserved quantities.

Both $T_1$ and $T_2$ are missing from our list that we can reconstruct so far, though $T_1$ is simply the horizontal impulse, and $T_2$ is the Hamiltonian.  Both of these forms require the Bernoulli equation which has not been considered other than via recursive substitution in $T_4$ and $T_6$.

In general, it would be nice to show that we could rearrange our general form along the lines of
$\displaystyle \int_S\left( \Psi^{(n)}(x,\eta,t) + t\phi(x,\eta)\frac{d}{dx}\Psi^{(n)}\right)\:dx = t\int_S\frac{d}{dt}\left(\phi(x,\eta,t)\frac{d}{dx}\Psi^{(n)}_x(x,\eta,t)\right)\:dx + J_\Gamma$

where
$\displaystyle \frac{d}{dt}\left(\phi(x,\eta,t)\frac{d}{dx}\Psi^{(n)}_x(x,\eta,t)\right)$ corresponds to some other conserved quantity in the hierarchy.  That is, can we relate this to lower order conserved quantities?