Example of reduction of number of parameters

Consider equation

\( \dot r = - (rv)' \) ; \( r \dot v = -Ar' - B v ^ S \)

Where \( r=r(x,t) \) and \( v=v(x,t) \) ; \(A\), \(B\), \(S\) are parameters (positive real numbers).
As usually, prime \( ' \) differentiates with respect to the first (primary) argument,
and dot \( \dot{} \) does the same with respect to the last argument (as dot appears at the end of the sentence).
Having no smart idea, I define

\( X=\alpha x \) ; \( T=\beta t \)

Let

\( R=R(X,T)=r(x,t) \) ; \( V=V(X,T)=\gamma v(x,t) \)

Then

\( R' = \alpha r' \) ; \( V'=\alpha\gamma v' \)

\( \dot R = \beta \dot r \) ; \( \dot V=\beta\gamma\dot v \)

\( \dot R = - \beta (R V/\gamma)' \) ; \( R \dot V \beta/\gamma = -(A/\alpha) R' - B (V/\gamma)^S \)

\( \dot R = - \frac{\beta}{\gamma} (R'V+ R V') \) ; \( R \dot V = - \frac{A \gamma}{\alpha\beta} R' - \frac{B}{\beta\gamma^{S-1}} V^{S} \)

I set \( \gamma=\beta \) ; then

\( \dot R = - R'V- R V' \) ; \( R \dot V = - \frac{A}{\alpha} R' - \frac{B}{\beta^{S}} V^{S} \)

I set \(\alpha=A\) and \( \beta=B^{1/S} \) ; then

\( \dot R = - R'V- R V' \) ; \( R \dot V = - R' - V^{S} \)

Compared to the initial equations, two parameters are excluded.
In this sense, the exact solution of the initial system of equations is acheived:

\( r(x,t) = R(x/A,\ t/\beta)\) ; \( v(x,t)=B^{-1/S}\ V(x/A,\ t/\beta) \)