Application of Lie groups and differentiable manifolds to general methods for simplifying systems of partial differential equations

General techniques are developed to obtain: (1) the completion of a systemof nonlinear first-order partid differential equations (PDES) which is an indepem dent set of further PDES derivable from the system by differentiation and elimination; and (2) simplifications of the system by choosing appropriate new independent and dependent variables using a result from Lie group theory The number of dependent and independent variables is reduced to the minimum. The theory specializes to the clasricd theory of a single nonlinear PDE with one unknown and can be combined with the methods of Olver, Edelen and Estabmok and Wahlquist. Most of the methods appear to be sufficiently well defined for automation as are the techniques in Olvcr. A second-order nonlinear equation in n dimensions is given which is related to a fuoctional differential equation in statistical mechanics. It is reducible to two dimensions for any value of n 2 2. 1. I n t r o d u c t i o n In this paper I will develop the idea of reduction of dimension for linear and nonlinear systems of partial differential equations (PDES). I t is an extension of Monge's method for tackling single PDEs of first order with one unknown. The method is applicable t o any system defined with sufficiently differentiable functions but the result is not usually one-dimensional; in fact there may be no reduction of dimension giving no simplification a t all. The result of the transformation is another system of PDEs having the same set of solutions with a possibly smaller number of independent variables but the number of dependent variables, which are the unknowns, may be increased initially but their number will afterwards be minimized. From the point of view of the general theory of systems of PDEs (called systems for brevity) the procedures indicated here should be applied initially, then symmetry methods should be applied if necessary. The best known of these are, firstly, looking for infinitesimal generators of geometrical symmetries [l] (isovectors of the differential ideal [2, 31) from which group invariant solutions can be obtained and the generalized method of characteristics ([4, 51). The latter method requires the initial da t a to satisfy an extra condition hut perhaps more flexibility can be obtained by applying the method to a prolongation of the original system (including derivatives of the dependent variables as new unknowns). Secondly there are the related methods of Estabrook and Wahlquist [6] originally applied to PDEs with two independent variables. They prolong the differential ideal 0305-4470/91/132913+29$03.50 @ 1991 IOP Publishing Ltd 2913