Solution
= Solution
The <chain rule> gives
$$
u_t=a f'(x+at),\qquad u_x=f'(x+at),\qquad
u_t+u_x=(a+1)f'(x+at).
$$
For this to vanish for every differentiable $f$, its coefficient must be zero:
$$
\boxed{a=-1.}
$$
The resulting $f(x-t)$ is a profile transported to the right at unit speed by the <transport equation>.