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 \varphi, we have the following relationship (modulo contributions from the lateral boundaries and the bottom):

\displaystyle \frac{d}{dt}\int\varphi\:dx = -\int\phi\frac{d\varphi_x}{dx}\:dx    (1)

Alternatively, the above can be written as

\displaystyle \frac{d}{dt}\int\varphi + t \Phi\:dx = t\int\frac{d}{dt}\left(\phi\frac{d\varphi_x}{dx}\right)\:dx    (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 \int g\eta\varphi_{xx} to be dealt with later.

If we’re considering \varphi to be of the form \varphi = (x + iz)^n, then the conditions that \varphi_{xz} = 0  and \varphi_{xx} 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 T_3, T_5 and T_6 but prevents us from creating a hierarchy as we can no longer use higher-degree powers of x + iz.

To find T_1, T_4, T_7 and T_8, we need to use a different strategy.

Finding the other Conservation Laws

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

\varphi_z\eta_t = \varphi_z\frac{d\phi}{dn}

we can replace the above form with

\displaystyle \int\varphi_z(\phi_z-\eta_x\phi_x)\:dx = -\int\phi\frac{d\varphi_x}{dx}\:dx    (3)

The downside is that we loose a t-derivative that hints at a conservation law.  The bonus is that can build new conservation forms from this.  Since \phi satisfies Laplace’s equation, we know that \phi_t will also satisfy Laplace’s equation.  Replacing $\phi$ with $\phi_t$ in (3), we find

\displaystyle \int \varphi_z\left(\phi_{tz}-\eta_x\phi_{tx}\right)\:dx = -\int\phi_t\frac{d\varphi_x}{dx}\:dx    (4)

This looks somewhat hopeless until you realize that we can rewrite the \phi_{zt} - \eta_x\phi_{xt} term as

\displaystyle \phi_{zt} - \eta_x\phi_{xt} = \frac{d}{dt}\left(\phi_z - \eta_x\phi_x\right) - \frac{d}{dx}\left(\eta_t\phi_x\right).   (5)

Likewise, we rewrite

\displaystyle \varphi_z\frac{d}{dt}\left(\phi_z - \eta_x\phi_x\right)  = \frac{d}{dt}\varphi_z\eta_{t} - \varphi_{zz}\eta_t^2  (6)

where here, I’ve substituted \eta_t = \phi_z - \eta_x\phi_x.  These chains of substitutions allow us to rewrite the above as

\displaystyle \frac{d}{dt}\int\phi\frac{\varphi_z}{dn}\:dx = \int \left(\phi_t\frac{\varphi_z}{dn} + \eta_t\phi_x\frac{d\varphi_z}{dx} + \varphi_{zz}\eta_t^2\right)\:dx    (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 \varphi = (x + iz)^2.  Then, adding

\displaystyle 2\frac{d}{dt}\phi\left(\varphi_{zz} + i\eta_x\varphi_{xz}\right)  (8)

to both sides of the equations and expanding the t derivative on the right-hand side will indeed yield a closed differential form provided that the degree of \varphi is 3 or less.  In fact, if the degree is exactly three, the integral on the left-hand side of (7) results in I^7 and I^8 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?

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