Trouble understanding method of characteristics-PDE for solving the Cauchy Problem So I am trying to understand the non-algorithmic part of the method of characteristics for solving a first order quasilinear PDE:

$ a(x,y,u)u_{x}+b(x,y,u)u_{y}=c(x,y,u) \hspace{1cm } $ (1)

I understand why the characteristic curves are solutions to the equation:

An integral surface $ z=u(x,y) $ has normal vector $(u_{x},u_{y},-1)$, so (1) is equal to the condition that the normal of the integral surface is perpendicular to the vector $(a(x,y,z),b(x,y,z),c(x,y,z))$.

So the characteristic curves are defined as the curves with tangent vector (a,b,c) for every point of the curve. I also understand how to obtain them solving the ODE's system.
The problem is that, in my notes, given the particular Cauchy problem:

$ \left\{\begin{matrix}
 a(x,y)u_{x}+b(x,y)u_{y}=c(x,y)  \\ 
 u(f(s),g(s))=h(s)
\end{matrix}\right.$

It is stated that the Cauchy problem has a solution if the curve $ \{f(s),g(s)\} $is not characteristic. I can't get why, Is this for the sole purpose of the change of variables to work well? I can neither understand why the characteristic curves are said to be "constant" solutions of the PDE.
 A: The idea of the method of characteristics is to change co-ordinates so as to rewrite the differential operator $(a \partial_x + b \partial_y)$ (the operator that is applied to $u$ on the left-hand side of your (1)) as a single derivative with respect to a certain variable. Let this variable be $t$, and this then gives you an ODE,
$$ \frac{d}{dt}u(x(t),y(t)) = c(x(t),y(t),u). \tag{2} $$
In particular, this equation holds on the characteristic curves, which are defined as you specify.
Provided that the characteristics only have point-like intersections with the initial data curve $(f(\xi),g(\xi))$, the inverse function theorem allows us to produce a parametrisation of the characteristics by where they intersect the initial data curve. 
Further, because (2) is an ODE on a characteristic, the values of $u$ on the characteristic only depend on the initial value of $u$ on the characteristic (i.e. the intersection of it with the $(f,g)$-curve) (as well as the values of $c$ along the characteristic), so this means that each point's initial data propagates along only one characteristic. If $c\equiv 0$, the data does not change along a characteristic, so the function is constant on each characteristic.
If you can do this, you can label every point where the initial data specifies values of $u$ by two co-ordinates: the characteristic variable $t$ (distance along a characteristic) and the characteristic it is on (given by intersection with the initial data at a specific $\xi$). With this change of coordinates, you re-express the function as a function of $(t,\xi)$, which gives you the system
$$ \partial_t u(t,\xi) = c(t,\xi,u) \\
u(0,\xi) = h(\xi), $$
which is easy to solve.
