The problem

Solutions of partial differential equations depend on arbitrary functions, represented, for example, by initial or boundary values, and since reality is (3 + 1)-dimensional (three spacelike and one timelike dimension), these functions will be functions of three variables if general solutions of physically important differential equations are being treated.

If we want to give these solutions in terms of “known” or even “simple” functions, or if we just want to find exact solutions, we practically always aim to express the solution in terms of functions of one (sometimes complex) variable. There are two main streams of dealing with partial differential equations and their solutions which are based on this experience.

The first is to try a separation ansatz, that is, to look for solutions that are products (or sums) of functions of one variable (and perhaps to give the general solutions in terms of such products). We shall come back to this problem in Chapter 20.

The second is to ask for solutions that depend on less variables than occur in the original formulation of the differential equation, maybe even on one variable alone. Typical examples are spherically symmetric solutions to the potential equation, or plane-fronted monochromatic waves in Maxwell theory.