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.*

That part I’m not happy with. However, along the way, there seems to be an idea worth pursuing. It’s straight-forward though tedious. The idea is as follows.

We know that for a Harmonic Function , we have the following relationship (modulo contributions from the lateral boundaries and the bottom):

(1)

Alternatively, the above can be written as

(2)

## Building Conservation Laws from (2)

Using (2), it’s easy to build conservation laws provided that the right-hand side of (2) comes from a closed form plus an additional contribution of the form to be dealt with later.

If we’re considering to be of the form , then the conditions that and is a constant restricts the power to be $2$ or smaller. These two constraints ensure that the we can fully integrate the equation.

This allows us to create the conserved quantities and but prevents us from creating a hierarchy as we can no longer use higher-degree powers of .

To find and , we need to use a different strategy.

## Finding the other Conservation Laws

Given that the left-hand side of (2) can be rewritten as

we can replace the above form with

(3)

The downside is that we loose a -derivative that hints at a conservation law. The bonus is that can build new conservation forms from this. Since satisfies Laplace’s equation, we know that will also satisfy Laplace’s equation. Replacing $\phi$ with $\phi_t$ in (3), we find

(4)

This looks somewhat hopeless until you realize that we can rewrite the term as

(5)

Likewise, we rewrite

(6)

where here, I’ve substituted . These chains of substitutions allow us to rewrite the above as

(7)

While this might seem even more hopeless, what’s fascinating is that the right-hand side is remarkably close to a closed differential form (again, modulo some “g” terms popping up from recursively substituting in the Bernoulli equation).

That is consider . Then, adding

(8)

to both sides of the equations and expanding the derivative on the right-hand side will indeed yield a closed differential form provided that the degree of is 3 or less. In fact, if the degree is exactly three, the integral on the left-hand side of (7) results in and on pg. 161 of Benjamin & Olver.

## A recursive pattern

Note that the above trick allowed us to extend our results to higher powers. Can we keep playing this trick?