For the conservation law
$\partial_tu+u\partial_xu=0$ for $(x,t)\in\mathbb{R}\times[0,T)$
$u(x,0)=\arctan(x)$ for $x\in\mathbb{R}$
I have been asked to find an explicit equation for $u(x,t)$. However I am having trouble doing so. I know that the critical time $T$ for which a classic solution exists is $\infty$ because $\partial_s(\arctan(s))=\frac{1}{1+s^2}>0$ for all $s\in\mathbb{R}$. (For a finite critical time we would need at least one point where $\partial_s(\arctan(s))<0$ which is not true).
So by the method of characteristics for $(X(\tau,s),T(\tau,s),Z(\tau,s))$ I get that $X=\arctan(s)\tau+s$, $T=\tau$, and $Z=\arctan(s)$. From this I know that $Z(X,T)=\frac{X-tan(Z(X,T))}{T}$, and so the implicit equation for $u(x,t)$ is $u(x,t)=\frac{x-tan(u(x,t))}{t}$. I have no idea how to get this into an explicit form for $u(x,t)$, though. (I already suspect that maybe my implicit function is wrong because of the discontinuity of $\tan$ but I'm not sure if that's relevant or not).