1.3 Formulation of the model in dimensionless form The dimensional analysis is performed essentially as in [3, 52]. In dimensional units, Z parametrizes the direction normal to the layers. The anode of the gas discharge is at Z $_{= 0}$, the cathode end of the discharge is at Z = $_{d$_{g}$}$, and the semiconductor extends up to Z = d$_{g}$ + $_{d$_{s}$}$. When diffusion is neglected, the ion current and the ion density at the anode vanish. This is described by the boundary condition on the anode Z $_{= 0}$: The boundary condition at the cathode, Z = $_{d$_{g}$}$, describes the $_{γ}$-process of secondary electron emission: Finally, a DC voltage U$_{t}$ is applied to the system determining the electric potential on the boundaries Here, the first potential vanishes due to gauge freedom. Let us introduce the intrinsic parameters of the system as t 0 = 1 α 0 µ$_{e}$E 0 , Z 0 = 1 α 0 , n 0 = ε 0 α 0 E 0 e . In the studies [3, 52], the problem was reduced to one spatial dimension z such that the GDS system takes the following dimensionless form, where the dimensionless time, coordinates and fields are z = Z Z 0 , τ = t t 0 , σ ( z, τ ) = n e ( Z, t ) n 0 , ρ ( z, τ ) = n$_{i}$ ( Z, t ) n 0 , E ( z, τ ) = E ( Z, t ) E 0 , φ ( z, τ ) = Φ ( Z, t ) E 0 Z 0 and α ( E ) = e − 1 / |E$^{|}$. The intrinsic dimensionless parameters of the gas discharge are the mobility ratio µ of electrons and ions and the length ratio L of discharge gap width and impact ionization length. That is, µ = µ$_{i}$ µ$_{e}$ and L = d$_{g}$ Z 0 . The boundary conditions become 7 ρ (0 , τ ) = 0 , σ ( L, τ ) = γµρ ( L, τ ) , φ (0 , τ ) = 0 , φ ( L, τ ) = −U , (1.17) ∂$_{τ}$ σ − ∂$_{z}$ ( E σ ) = σ E α ( E ) , ∂$_{τ}$ ρ + µ∂$_{z}$ ( E ρ ) = σ E α ( E ) , ∂$_{z}$ E = ρ − σ, E = − ∂$_{z}$φ, (1.16) Φ (0 , t ) = 0 , Φ ( d$_{g}$, t ) = − U. (1.15) | Γ$_{e}$ ( d$_{g}$, t ) | = γ | Γ $_{i}$( d$_{g}$, t ) | ⇒ µ e n e ( d$_{g}$, t ) = γµ$_{i}$n$_{i}$ ( d$_{g}$, t ) . (1.14) Γ $_{i}$(0 , t ) = 0 ⇒ n$_{i}$ (0 , t ) = 0 . (1.13)