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 where
He also suggested looking at these quantities for the stream function as well. Given that if is harmonic, so is , I replaced with in the nonlocal equation. This results is a more complicated expression given by (modulo a sign error or two)
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 ). This also gives a very straightforward way to incorporate the Bernoulli equation through the 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.