= Solution
For
$$
p(\rho)=A-\frac{B^2}{\rho},
$$
the <adiabatic sound speed> is
$$
c^2=\frac{dp}{d\rho}=\frac{B^2}{\rho^2},
\qquad c=\frac B\rho.
$$
Since $\int(c/\rho)d\rho=-B/\rho=-c$, choose invariants that vanish in the equilibrium state $(u,c)=(0,c_0)$:
$$
\boxed{R_+=u-c+c_0,
\qquad R_-=u+c-c_0}.
$$
Consequently
$$
\boxed{u+c=c_0+R_-,
\qquad u-c=-c_0+R_+}.
$$
In a right-moving <simple wave> with $R_-=0$, the $R_+$ characteristic speed is therefore $u+c=c_0$, independent of amplitude. The transport equation becomes $(\partial_t+c_0\partial_x)R_+=0$, so the whole velocity profile translates rigidly at the equilibrium <sound speed> $c_0$ without nonlinear distortion.
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