I tried with DSolve and NDSolve, but Mathematica fails.
With Maple 2020 I have:
Translating to Mathematica code:
ClearAll["`*"]
sol = {R[z] -> R0*Exp[((Log[U0/UL] + 2*I*Pi*n)*z)/(2 L)],
u[z] -> U0/
Exp[((Log[U0/UL] + 2*I*Pi*n)*z)/(2 L)]^2};(* n\[Element]Integers *)
u[z] /. sol[[2]] /.
z -> 0(*The equation satisfy a initial condition u[0]=U0 *)
(* U0 *)
{sol[[1]] /. n -> 0, sol[[2]] /. n -> 0}(*for n=0*)
(* {R[z] -> R0 (U0/UL)^(z/(2 L)), u[z] -> U0 (U0/UL)^(-(z/L))} *)