Nonlocal symmetries of evolution equations

TL;DR

This paper introduces a method for classifying evolution equations with nonlocal symmetries, leading to the discovery of new nonlinear equations beyond classical Lie symmetry approaches.

Contribution

The paper develops a novel classification method for evolution equations with nonlocal symmetries, expanding the scope of symmetry analysis beyond traditional Lie methods.

Findings

Constructed broad families of nonlinear evolution equations with nonlocal symmetries

Identified equations invariant under Lie algebras of dimension up to three

Demonstrated equations cannot be obtained via classical Lie symmetry methods

Abstract

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.