Solutions to Nonlinear Partial Differential Equations from Symmetry-Enhancing and Symmetry-Preserving Constraints

We show how solutions to practical partial differential equations can be found by classical symmetry reductions of a larger system of equations including constraints that result in the enlarged system having a larger symmetry group or an identical symmetry group compared to the original target equation. Symmetry-enhancing constraints provide additional similarity solutions beyond those of the standard Lie algorithm for scalar equations. Symmetry-preserving constraints enable solutions to be found more easily, after reduction of variables. Examples are given for the cylindrical boundary layer equations and for a Navier]Stokes formulation of Schr¨odinger wave mechanics