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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.

Dimitri_896
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    In which functional space are you working so that you have this "identity" ? This seems ill-defined as $g$ could be 0 as far as I can tell, plus what are the $x_i$? – Flewer47 Mar 01 '21 at 10:54
  • Functional space, what? This is a standard delta function identity which holds around the zeroes of a function $x_i$. I'm simply asking what happens when $g(x)$ is a prescribed function. – Dimitri_896 Mar 01 '21 at 11:17

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