So I know we have this "identity" $$ \delta(g(x)) = \sum_{i} \frac{\delta(x-x_i)}{|g(x)'|}\\ = \frac{1}{|g(x)'|} $$
What about when $g(x)$ is a given function, say the simple wave solution to the Hopf Equation, i.e.: $$ g(x) = u-u_0(x-ut)$$
This gets me
$$\frac{1}{|1+\frac{du}{dx}\cdot t|}$$
I appreciate this is an unorthodox and unusual question but I'm curious to see how this would change the result.