Reductions of Differential Equations: Scaling Symmetries

As an example of Lie's technique, consider the standard linear second-order ordinary differential equation that we discuss in sophomore differential equations,

If we rescale by an arbitrary constant , making the change of variable , we are left with exactly the same equation,

We say in this case that equation (1) is scale invariant and that we have identified a scaling symmetry of the equation.

In order to apply this scale symmetry to help us solve equation (1), we rewrite the equation as a pair of first-order equations,

and then ask: "Are there any quantities that are left invariant by the scaling symmetry?" Computation of such an invariant is algorithmic, but we will merely notice that is identical in both the original and in the new coordinate system. Assuming that is a function of , we next write the differential equations that must satisfy. We do this by replacing with the product in equations (2) and (3) to get

and

This implies that

We have converted the original problem to two integrals, namely

and, once is known,

However, we recognize that we can actually say a lot about equation (5). Namely, it is the Ricatti equation that corresponds to equation (1). Equation (4) is the Ricatti transformation. Furthermore, if we choose the two special solutions to equation (5) given by the roots of the right-hand side and then solve equation (4) for , we have recovered the standard technique for solving linear constant coefficient differential equations that is taught in the sophomore course.

Our second example is the heat equation in one spatial dimension,

It is preserved by the change of variables given by the scalings , , and . If we again search for quantities that are invariant under these scalings, we discover the new variables and where satisfies . We assume that is a function of and compute the equation for that arises by insisting that be a solution of the heat equation. Setting and substituting into equation (6) yields an ordinary differential equation for ,

This equation is Kummer's equation and the solution of it may be expressed in terms of confluent hypergeometric functions, and . Mathematica knows these two functions as Hypergeometric1F1[] and HypergeometricU[]. We can use solutions to Kummer's equation to plot scale invariant solutions of the heat equation. Figure 1 is the result of setting . It was generated with the following routine.

Figure 1. A scale invariant solution of the heat equation.

The horizontal axis gives the space variable and the vertical axis is . White sections are high temperature regions while dark regions are cooler.