Cannot understand method of characteristics $u_t+uu_x =0$
$u(x,0)\equiv u_0(x)=\begin{cases}
 0 & x<0 \\
 1 & x>0
\end{cases}$
I want to parametrise $u(x(s),t(s))$. This is the first thing that is conceptually quite difficult to picture, but I get the idea that we are parameterising.
$$\frac{du}{ds}=\frac{\partial u}{\partial x}\frac{dx}{ds} +\frac{\partial u}{\partial t}\frac{dt}{ds}$$
In this case
$$\frac{du}{ds} = 0$$
So I understand this is just a trick to use and so
$\frac{dx}{ds}=u$, $\frac{dt}{ds}=1$.
Next $\frac{dx}{dt}=u$, do we always do this, i.e. do we always divide those derivatives to eliminate the parameter? If I do then I lose it forever, and it seems that the constant I get should actually be the parameter, $s$, that is why it is used strangely below, I cannot seem to work out how to do this properly.
$x=ut+\text{const.}\equiv ut+c$
If $t=0$ then $x=c$ and $u(x,0)=u_0(c)=\begin{cases}
 0 & c<0 \\
 1 & c>0
\end{cases}$
Is $x=u_0t+c$ equivalent? *
Then $x=\begin{cases}
 c & c<0 \\
 t+c & c>0
\end{cases}$
This should read 
$x=\begin{cases}
 s & s<0 \\
 t+s & s>0
\end{cases}$
What have I misinterpreted here?
*Edit: On the Wikipedia article http://en.wikipedia.org/wiki/Method_of_characteristics#Example it uses the fact that $u_s=0$; the solution is constant along the characteristic - then that $(x_s,t_s)$ and $(x_0,0)$ are on the same characteristic (How?) to deduce $u(x_s,t_s)=u(x_0,0)$ where $(x_s,t_s) = (a,1) $.
This would seem to be a solution to taking $u=u_0$ in my (above) attempt
** In another attempt, after reading the Wikipedia page,
$t_s=1$, $x_s=u \left(\equiv u(x(s),t(s))\right)$ and $u_s=0$ as before,
$\int_{t_0}^t dt=\int_{0}^s ds \Rightarrow t=s+t_0$ and since $t_0=0$ (Why?) $t=s$. 
$x_s=u$ I am guessing that all characteristics must originate from some initial point in the the initial value problem. So for this reason, $u(x,0)=u(x(s),t(s))$ since $u$ is constant on the characteristic. 
This leaves me with $\int_{x_0}^x dx=\int_0^s u_0 ds$ which leads me to $x=u_0 t+x_0$ since $u$ is constant along the path of $s$ and is therefore a constant in that integral, and I replace $s$ with $t$. The only way to get to the answer from here is with $x_0=s$ - I cannot see how you would get here, since $t=s$.
 A: There is no particular reason to parametrize $t$, and it is simpler not to. 
From the chain rule, along a curve $(x(t),t)$, you have 
$$
 \frac{d}{dt}(u(x(t),t) = u_t+x'(t)u_x,
$$
and set this equal to $u_t+uu_x=0$,
from which $u$ is constant on lines where $x'=u$. In your case, all vertical lines
starting from $(x,0)$ when $x$ is negative, so $u=0$ in the whole left half plane.
Also all lines of slope 1 starting from $(x,0)$ when $x$ is positive, so $u=1$
in that whole region. That leaves a wedge shape where $u$ has not been defined yet.
The only way a line can fit into that is if it is $x=mt$ with $m$ between 0 and 1.
That gives you a solution $u=x/t$ in that wedge, and together with the first two
regions, now $u$ is defined and continuous in the whole half space $t>0$.
You might parametrize both $t$ and $x$ for an equation like 
$a(x,t,u)u_t+b(x,t,u)u_x=0$, but not needed here.
A: I think you are thinking this question too complicated.
Just follow the following procedure is OK!
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\dfrac{du}{ds}=0$ , letting $u(0)=u_0$ , we have $u=u_0$
$\dfrac{dx}{ds}=u=u_0$ , letting $x(0)=f(u_0)$ , we have $x=u_0s+f(u_0)=ut+f(u)$ , i.e. $u=F(x-ut)$
$u(x,0)=\begin{cases}0&x<0\\1&x>0\end{cases}=H(x)$ :
$F(x)=H(x)$
$\therefore u=H(x-ut)=\begin{cases}0&x-ut<0\\1&x-ut>0\end{cases}=\begin{cases}0&x<0\\1&x-t>0\end{cases}=\begin{cases}0&x<0\\1&x>t\end{cases}$
Hence $u(x,t)=\begin{cases}0&x<0\\1&x>t\\c&\text{neither}~x<0~\text{nor}~x>t\end{cases}$
