I am trying to solve burgers equation with initial value $u(x,0)= arctan(x)$. Fixed $x_0$, I wrote the Ode for finding the characteristic: $\frac{dx}{dt}=u(x(t),t)$. Now let's define $z(t)=u(x(t),t)$. I know that $z'(t)=0$ for the equation itself so I get that $z(t)=z(0)= u_0(x_0)=arctan(x_0) $.
Replacing it into my ode I get $x(t)=(arctan(x_0))t+x_0 $ which I know is invertible and I should get an expression for $x_0$ to replace in $u_0$ to get the solution. Am I wrong? How can i get the solution?