Non-invertible mappings of linear PDEs to nonlinear PDEs through the symmetry-based method

Abstract

We show that the well-known Hopf–Cole transformation mapping the linear heat equation to the nonlinear Burgers' equation naturally extends to the mapping of any linear PDE to a non-invertibly equivalent nonlinear PDE. This mapping is obtained through the symmetry-based method by using the admitted obvious scaling symmetry in the dependent variable of any linear homogeneous PDE. Furthermore, each nontrivial point symmetry of any linear PDE yields a corresponding nonlocally related nonlinear PDE. The mapping relating the linear PDE and the corresponding nonlinear PDE is not one-to-one. As examples we consider the linear heat equation, the linear wave equation, Laplace's equation and the Helmholtz equation in two or more independent variables. We exhibit some exact solutions of the corresponding nonlinear system of PDEs from a known solution of the associated linear PDE. Moreover, we find nonlocal symmetries for the corresponding nonlocally related nonlinear systems of PDEs through the commutator relationship between point symmetries of the associated linear PDE.