The author treats the problem of the complete specification of the Monge–Ampère equation in terms of its Lie point symmetries. As an initial step a simple translation of the dependent variable reduces the Monge–Ampère equation for some constraint upon its parameters to \[ u_{tt}u_{xx}-u^2_{tx}=\kappa\tag{1} \] which is known as the Bateman equation when \(\kappa=0\). (The Bateman equation arises as a condition in the singularity analysis of certain nonlinear partial differential equations.)
The Lie point symmetries of (1), nine in all, are calculated and from there it is shown that \[ \Delta(t, x, u, u_t , u_x , u_{tt} , u_{tx} , u_{xx}) = 0 \] reduces to (1) after the imposition of invariance under the action of these symmetries. Equation (1.1) is described as ‘Lie-remarquable’.
Equations in more variables are similarly treated. Then Monge–Ampère equations of higher order are considered. A variation of the translation of the dependent variable mentioned above produces considerable simplification. In the case of the third-order equation the Lie algebra is of dimension ten for general values of the parameters, but with a further constraint there are eleven symmetries and these are sufficient to specify this precise equation completely.
Similar considerations apply at higher orders.
This paper is an interesting study of the complete symmetry group of a partial differential equation nearly two centuries old and yet still in popular usage. It appears that the author is unfamiliar with recent literature on complete symmetry groups, but is more used to concepts developed in a separate, closely related, line of thought.
For the entire collection see [Zbl 1089.35008].