Method of characteristics for 2x2 systems In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE systems.
Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this:
$$ \begin{cases}
u_t+f(u,v) u_x=h(u,v) \\[2ex] 
v_t+g(u,v) v_x=m(u,v) \\[2ex]
(u(x,0),v(x,0))=(u_0 (x), v_0 (x))
\end{cases}
$$
(This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. 
$$(u(x,0),v(x,0))= \begin{cases}
 (u_l , v_l), x<0 \\[2ex]
 (u_r , v_r), x\geq0
\end{cases}$$
we would talk about Riemann problem.
I want to solve this problem using method of characteristics.
My question is: Is there any book/note/paper where I could find a step by step solution of quasilinear system that looks like this by using method of characteristics? Or maybe an easier quasilinear system (e.g. if we put right hand sides equal zero, put $f$ equal to $g$,...). It doesn't matter if it is a Riemann or Cauchy problem. I just want to see how system of this type could be solved in detail from start to finish. It could be shown on any concrete system (and not for the general case I've written above).
In the literature I've found something in the book: 
Toro - Riemann solvers and numerical methods of fluid dynamics, 2009, 
but it is very short. Additionaly, I have found a similar questions on Mathoverflow (question1 and question2).
If anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine. 
 A: When the RHS is 0, you are basically asking about the method of Riemann invariants. A quick summary of the method you can find in the second section of 
Lax (1964), J. Math. Phys.
Alternatively, I am pretty sure it is also discussed somewhere in the second volume of Courant and Hilbert. 
EDIT: I see that there's also a decent write-up on Wikipedia, you just need to know where to look. 
A: The method of characteristics for $2\times2$ systems is discussed by C. Dafermos in his book. In the third edition, it is Chapter XII.
However, I want to make a few points:


*

*as mentionned by Willie, it was elaborated by P. Lax, who used it to prove that genuine nonlinearity implies the blow-up of first derivatives in finite time, for very general initial data. 

*The method is of little help for applications, because we don't want to be llimited by the onset of shock waves.

*Lax calculus has been generalized to $n\times n$ systems of the rich class. A system is rich if it can be put in a conservative form and it can also be put in a diagonal form. This is called semi-hamiltonian system by the russian school.

