I am trying to confirm a conservation law I cam across in a paper (Janssen 1983 "On a fourth-order envelope equation for deep-water waves" Journal Fluid Mechanics), and am having difficulty.
In particular, I'm trying to confirm the conservation of linear momentum $P$, where
$P = \frac{i}{2} \int AA^*_x - A^*A_x \ dx$,
where $^*$ denotes complex conjugate and spatial integrals are taken to be over all space in this question. Now, the governing equation takes the form
$A_t = \mathcal{N}(A,A^*) - P.V. \ i\alpha\ A\int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$,
where $\mathcal{N}$ is a (nonlinear) operator describing the rest of the dynamics, which is not important for this question, P.V. denotes the principal value and $\alpha$ is some real constant. Note, this integral is proportional to the Hilbert transform.
Now, I want to look at the time evolution of the momentum $P$. To that end, we have
$\frac{d P}{dt} = \frac{i}{2} \int \dot{A}A^*_x +A\dot{A}_x^* -\dot{A}^*A_x -A^*\dot{A}_x \ dx$
Substituting in the relation for our governing equation, and using integration by parts (we assume the field A is compact) we find
$\frac{dP}{dt}=\frac{\alpha}{2} \int (|A|^2)_x \mathcal{H} (|A|^2_x) \ dx$
where
$\mathcal{H}(|A|^2_x) \equiv P.V. \int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$.
Now, this term is non-zero, where as the author claims this integral is conserved. I don't see why this integral should vanish, but perhaps I'm not exploiting a property of the Hilbert transform.
Am I missing something obvious?
Thanks,
Nick
