How symmetries yield non-invertible mappings of linear partial differential equations

Abstract

Nonlocally related partial differential equation (PDE) systems are important in the analysis of a given PDE system. Useful nonlocally related systems can be constructed through conservation law and symmetry based methods. In this paper, we focus on an application of the symmetry-based method to linear PDE systems. In particular, we show how to obtain systematically non-invertible mappings of linear PDEs to linear PDEs. As examples, we obtain non-invertible mappings of the Kolmogorov equation with variable coefficients to the backward heat equation (a PDE with constant coefficients) as well as non-invertible mappings of linear hyperbolic PDEs with variable coefficients to linear hyperbolic PDEs with constant coefficients.