How to proceed with dimensionless recasting of telegrapher's equation?

I am referring to this paper, on page 3 it is given that:

The telegrapher's equation is given as $$\partial_t P_+=D\partial_x^2P_+-v\partial_xP_+-\gamma P_++\gamma P_-,$$ $$\partial_t P_-=D\partial_x^2P_-+v\partial_xP_-+\gamma P_+-\gamma P_-.$$ where $P_+(x; t)$ and $P_-(x; t)$ are the probability density for the particle to be at position x with velocities $+v$ and $-v$, respectively. Furthermore, choosing $\gamma^-1$ as the unit of time and $v\gamma^-1$ as the unit of length to recast the above given equations in the dimensionless form, we get $$\partial_t P_+=\mathcal{D}\partial_x^2P_+-\partial_xP_+- P_++P_-,$$ $$\partial_t P_-=\mathcal{D}\partial_x^2P_-+\partial_xP_-+ P_+- P_-$$ where $\mathcal{D}=D\gamma/v^2$.

This transition is not a simple subsitution, I want to know how they they have proceeded with recasting equations to dimensionless form. I tried to check in wikipedia, but couldn't find it.

1 Answer

I am able to answer the question using help from wikipedia link: taking $P_+=aP_+$,$P_-=bP_-$, $t=t_c\tau$, $x=x_c\zeta$, where $\tau , \zeta$ are the new independent variables. We can solve the given equation, for dimensionality take $\gamma/v=1/x_c$, $b/a=1$ and $t_c\gamma=1$ all these are dimensionally compatible leading to the given normalization. :)