I am considering the Fisher-KPP equation in the following form:
$$ \frac{\partial u }{\partial t'} = D \frac{\partial^2 u}{ \partial x'^2} + \rho u(1-u) $$
Now, with the nondimensional variables:
$$ t= \rho t',\\ x = \sqrt{\frac{\rho}{D}}x' $$ I can write the equation in nondimensionalized form: $$ \frac{\partial u }{\partial t} = \Delta u + u(1-u) $$ I think I understand it until here. The Fisher-KPP equation allows for traveling wave solutions, so that it can also be written in co-moving form:
$$ 0 = \Delta u +u(1-u) - v\nabla u $$ Is it possible to introduce a nondimensionalization that directly leads to this comoving formulation? All the material I could find merely re-scales the variables by a reference quantity, but I have not found anything with co-moving transformations. I tried to use $x = x' -vt$, but v does not even show up in the original formulation. What is the clean way to do this?
