We are going to discuss several topics involving hyperbolic PDEs. We will focus on the so-called characteristic curve. We start with the following first-order equations
with the init data
Advection Equation. In this section, we consider the case when is a constant. We show that the initial value problem for the advection equation, namely
has a unique solution given by , which is a right traveling wave of speed .
The way to see it is to make a change of coordinates to
Replacing all and derivatives by and derivatives. By the chain rule
The equation now takes the form in the new (primed) variable. Thus the solution is
with an arbitrary function of one variable. The initial condition (at ) tells us that . Thus
Characteristic curves. The above solutions suggests that is constant on the curve where is constant. The totality of all the curves , where is constant, is called the set of characteristic curves for this problem.
The constant acts as a parameter and distinguishes the curves. A graph of the set of characteristic curves on a spacetime diagram (i.e., in the plane) is called the characteristic diagram for the problem.
The characteristic curves, or just characteristics. are curves in spacetime along which signals propagate. In the present case. the signal is the constancy of that is carried along the characteristics. Further-and this will turn out to be a defining feature-the PDE reduces to an ordinary differential equation along the characteristics.
We end this section by proving an example.
The equation is with the initial data . Thus the general solution is .
This picture shows that in the case when coefficient is constant, the characteristic curves are parallel. This is not true in the case when is dependent of either or . This circumstance is discussed here.