On the Origins of Symmetries of Partial Differential Equations: the Example of the Korteweg—de Vries Equation

Abstract

Type II hidden symmetries of partial differential equations (PDE) are extra symmetries in addition to the inherited symmetries of the differential equations which arise when the number of independent and dependent variables is reduced by a Lie point symmetry. (Type I hidden symmetries arise in the increase of number of variables.) Unlike the case of ordinary differential equations, these symmetries do not arise from contact symmetries or nonlocal symmetries. In fact, we have previously shown that they are symmetries of other differential equations. However, in determining the origin of these symmetries we show that finding the origin of any symmetry of a PDE is a non-trivial exercise. The example of the Korteweg-de Vries equation is used to illustrate this point.