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

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 Schrödinger 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.

Latex2e, 14 pages, no figures; final form

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• Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case
• Galilean conformal algebras in two spatial dimension
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• Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case