Let the system: $$\alpha(x,t,u)u_t+\beta(x,t,u)u_x=f(x,t,u)$$ To find the characteristic equations: $$\frac{du}{ds}=\frac{\partial{u}}{\partial{t}} \frac{dt}{ds}+\frac{\partial{u}}{\partial{t}} \frac{dx}{ds}$$ $$u=u(x(s),t(s))$$
The characteristic system for first degree equation: $$ds=\frac{dt}{\alpha(x,t,u)}=\frac{dx}{\beta(x,t,u)}=\frac{du}{f(x,t,u)}$$
We have reduced the solution of a system of PDEs to the solution of a system of ODEs. $$$$ Could you explain me why we have found the characteristic system? Do we need this to solve hyperbolic equations?
The characteristic system shows you how the information (initial conditions) propagates through space and time. This is great for problems such as seeing how shock waves move. It is also leads to a nice way to generate numerical boundary conditions for a truncated domain (Riemann Invariants).