ABSTRACT
This chapter considers nonlinear PDEs of the general smooth form. The two problems solved in this chapter are to use parametrisation for defining globally arbitrary Lie group actions both on classical and generalised functions, some of them solutions of nonlinear PDEs; and to prove the corresponding basic property, namely, that even in the parametric case, usual Lie group symmetries of the PDEs will transform their solutions into solutions, both in the classical and in the generalised case. The parametric method amounts to an embedding of the usual definition of a function into a larger concept. The idea to dispense with our usual attachment to the distinction between independent and dependent variables was suggested in a different context. The class of such Lie semigroup actions is far larger than that of Lie group actions.
