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.

Darboux' Method(in French,la methode de Darbouxand so rendered in English asthe method of Darbouxin many places). However, the generic system is not explicitly solvable by Darboux' Method, as was shown by Lie in particular cases, with the general theorem proved by Goursat some years later (but, in any case, by 1896). $\endgroup$ – Robert Bryant Dec 17 '18 at 12:50