@robertocooper I missed the point that the eqn for v was to be derived from the imaginary part, and have added an edited version transform4.mw to my answer above. In your second example, you divided by exp(I*mu), but that left you with a mix of old and new variables. I tidied it up a bit. (I usually leave off the =0 in eqns since dsolve, solve etc understand this as default, and then I do not have to worry whether various simplifications apply automatically to both sides of an eqn or not.)
For the subs, I'm not sure if you are asking a mathematical or Maple question. For the mathematical question, abs(z) is the magnitude of the complex quantity z=r*exp(I*theta), i.e., abs(z)=r. From the Maple point of view, using subs does a substitution without doing any simplification, i.e., just syntactic substitution. I didn't want any attempt to try to apply diff to abs. More usually I would used eval(expr,a=b) rather than subs(a=b,expr) since it is smarter about math things. But here eval messed with the PDE too much.