Rigorous explanation of the method of characteristics for (the Cauchy problem) for first order PDEs Where can I find a rigorous explanation of the method of characteristics for  solving first order PDEs?
I'm particularly interested in the Cauchy problem associated to equations of the kind $$\partial_t u + a(t,x,u) \partial_x u = b(t,x,u),$$
where $u:[0,\infty)\times \mathbb{R} \to \mathbb{R}$.
 A: The equation you are interested in, is called a quasi-linear PDE, since it is linear in the first order derivative of $u$. The basic idea of the method consists in considering the rato of change of $u(x(t),t)$ from a moving position given by $x=x(t)$. By applying the chain rule we can write:
\begin{equation}
\frac{d}{dt}u(x(t),t)=u_t(x(t),t)+u_x(x(t),t)x^{'}(t)
\end{equation}
The first term on the right-hand side of the above above equation represents the change of $u$ at a fixed point $x$, while the second term  on the right-hand side is the change of $u$ due to the movement of the observation position. We can now compare the above equation to your quasi-linear PDE .
\begin{equation}
\partial_t u + a(t,x,u) \partial_x u = b(t,x,u)
\end{equation}
If we assume that $x^{'}(t)=a(x(t),t)$ from the PDE we see that:
\begin{equation}
\frac{d}{dt}u(x(t),t)=u_t(x(t),t)+a(x(t),t) u_x(x(t),t) = b(x(t),t)
\end{equation}
By integrating $x^{'}(t)=a(x(t),t)$ it is possible to determine the position of the point $x$ with respect to $t$. Thus:
\begin{equation}
x(t)=\int{a(t) dt}+x_0
\end{equation}
This formula defines a family of curves in the (x,t)-plane, called
characteristics.
Usually the PDE is associated to an Initial Value Problem (IVP) such as with  $u(x,0)=f(x)$. To find the value of the solution $u$ at $(x,t)$, we consider the characteristics of the equation $x(t)=\int{a(t) dt}+x_0$ , which intersects the x-axis. Since u is constant on the characteristics we can write
\begin{equation}
u(x,t)=u(x_0,0)=f(x_0)
\end{equation}
By replacing $x_0$ we have:
\begin{equation}
u(x,t)=u(x_0,0)=f(x(t)-\int{a(t) dt})
\end{equation}
where
\begin{equation}
\frac{d}{dt}u(t)= b(t)
\end{equation}
The method of characteristic is not limited to the IVP, it can be adapted to more general cases.
Concerning books for deeper explanations you can look at "Partial Differential Equations: a Genuine Introduction to" by Bode Vladimov .
A: In my opinion, having taught PDEs several times, the best and most convincing presentation of the method of characteristics appears in Fritz John's "PDEs".
Although, it is a graduate textbook, understanding the presentation of the method of characteristics does not require any prerequisites other that the knowledge of existence and uniqueness of solutions of systems of ODEs with smooth flux.
