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## 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 $_{= 0}$: 

<img src='content_image/316892.jpg'>

The boundary condition at the cathode, Z = $_{d$_{g}$}$, describes the $_{γ}$-process of secondary electron emission: 

<img src='content_image/316891.jpg'>

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. 

<img src='content_image/316890.jpg'>

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, 

<img src='content_image/316889.jpg'>

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 

<img src='content_image/316888.jpg'>

