Eikonal Equation - Mathematical Description

Mathematical Description

An eikonal equation is one of the form

The plane can be thought of as the initial condition, by thinking of as We could also solve the equation on a subset of this plane, or on a curved surface, with obvious modifications. This shows up in geometrical optics for example, where the equation is . There it is an equation describing the phase fronts of waves. Under reasonable hypothesis on the "initial" data, the eikonal equation admits a local solution, but a global solution (e.g. a solution for all time in the geometrical optics case) is not possible. The reason is that caustics may develop. In the geometrical optics case, this means that wavefronts cross.

We can solve the eikonal equation using the method of characteristics. Note though that one must make the "non-characteristic" hypothesis for We must also assume, for

First, solve the problem, . This is done by defining curves (and values of on those curves) as

Note that even before we have a solution, we know for due to our equation for .

That these equations have a solution for some interval follows from standard ODE theorems (using the non-characteristic hypothesis). These curves fill out an open set around the plane . Thus the curves define the value of in an open set about our initial plane. Once defined as such it is easy to see using the chain rule that, and therefore along these curves.

We want our solution to satisfy, or more specifically, for every, Assuming for a minute that this is possible, for any solution we must have

,

and therefore

In other words, the solution will be given in a neighborhood of the initial plane by an explicit equation. However, since the different paths, starting from different initial points may cross, the solution may become multi-valued, at which point we have developed caustics. We also have (even before showing that is a solution)

It remains to show that, which we have defined in a neighborhood of our initial plane, is the gradient of some function . This will follow if we show that the vector field is curl free. Consider the first term in the definition of . This term, is curl free as it is the gradient of a function. As for the other term, we note

The result follows