When Nonlinear Differential Equations are Equivalent to Linear Differential Equations

Abstract

Abstract. A necessary and sufficient condition is established for the existence of a 1-1 transformation of a system of nonlinear differential equations to a system of linear equations. The obtained theorems enable one to construct such transformations from the invariance groups of differential equations. The hodograph transformation, the Legendre transformation and Lie’s transformation of the Monge-AmpSre equation are shown to be special cases. Noninvertible transformations are also considered. Examples include Burgers ’ equation, a nonlinear diffusion equation and the Liouville equation. Introduction. In this work, we study transformations mapping nonlinear differen-tial equations to linear differential equations in a 1-1 manner. Based upon the group analysis of differential equations, we obtain necessary and sufficient conditions for the existence of such transformations. The established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct