On nonlinear partial differential equations with an infinite-dimensional conditional symmetry

TL;DR

This paper investigates the invariance properties of nonlinear PDEs under an infinite-dimensional Lie algebra, revealing the absence of second-order invariant equations but identifying a class of strongly nonlinear conditionally invariant PDEs with new solutions.

Contribution

It introduces a new class of strongly nonlinear PDEs invariant under an infinite-dimensional algebra, linking them to the Monge-Ampere equation and providing new exact solutions.

Findings

No second-order invariant equations under massless realizations.

A class of strongly nonlinear conditionally invariant PDEs identified.

New exact solutions for some PDEs in this class.

Abstract

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra A_N(z) in N spatial dimensions is studied. The special case A_1(2) was introduced in J. Stat. Phys. {\bf 75}, 1023 (1994) and contains the Schr\"odinger Lie algebra sch_1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A_N(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A_N(z) such that the well-known Monge-Ampere equation is the required additional condition. New exact solutions are found for some representatives of this class.