Non-local Symmetries of Nonlinear Field Equations: an Algebraic Approach

TL;DR

This paper introduces an algebraic method to identify non-local symmetries in nonlinear field equations, exemplified on the Dym and KdV equations, revealing a general formula for their infinitesimal operators.

Contribution

The paper develops a novel algebraic approach using infinite-dimensional subalgebras to find non-local symmetries of nonlinear equations, applicable to various equations with complex prolongations.

Findings

Derived a general formula for non-local symmetry infinitesimal operators.

Applied the method to Dym and KdV equations successfully.

Potential to analyze other nonlinear equations with nontrivial prolongations.

Abstract

An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra $L$ associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equations, allows us to obtain a general formula for the infinitesimal operator of the non-local symmetries expressed in terms of elements of $L$. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations.