PDEs
Given the hyperbolic PDE given by the wave equation:
with initial conditions and .
In 1D, it is easy to see that for a given solution , the change of variables and , provide another solution:
which can be visualized in the -plane as the shape of the initial wave moving with the speed (slope) . These lines represent propagation of information by a hyperbolic PDE.
More generally, this change of variables transforms the PDE into the separable, homogenous PDE:
where now the solution can have different “shapes” in the two propagation directions:
What we have done is identify that along the lines the PDE becomes an ODE. Generally, if is a function of , we want to know how changes w.r.t. , given by the total derivative:
If and then we arrive at the transport equation, where the particle moves at constant speed. Importantly, in this case, the function does not change along the line defined by . So, if we know then we know it for all times. These lines are called the characteristic lines of the PDE.
Now, for a nonlinear PDE, the characteristics can be nonlinear as well, i.e. curves. The question now becomes, how can we find these curves, given some nonlinear PDE? The simplest change we can make to the above to get a nonlinear PDE is to set . If we keep , this is called the inviscid Burger’s equation: