How can we check by direct differentiation that the formula $u(x, t) = \varphi(z)$, where $z$ is given implicitly by $x − z = ta(\varphi(z))$, does indeed provide a solution of the PDE $u_t + a(u)u_x = 0$?
So here's my intuition:
find value of $z$ from $x − z = ta(\varphi(z))$, by isolating it and then substitute; I only can think of characteristics, but really can't see about the a(u)