Characteristic solution of a scalar conservation law ($u(t,X_t(\xi ))=u_0(\xi )$)

Before characteristic crossing, the scalar conservation law $u_t+f(u{)}_x=0$ has solution $u(t,X_t(\xi ))=u_0(\xi )$ with $X_t(\xi )=\xi +t{f}^{\prime }(u_0(\xi ))$. When this map is a diffeomorphism, the solution is $u_0\circ X_t^{-1}$. Its spatial derivative is $u_0^{\prime }(\xi )/(1+t{f}^{\prime \prime }(u_0(\xi ))u_0^{\prime }(\xi ))$.