What question did this study set out to answer?

The research aims to classify nonlinear evolution equations that allow quasi-local symmetries through an algebraic framework.

March 29, 2026Open Access

On a Lie Group Characterization of Quasi-Local Symmetries of Nonlinear Evolution Equations

Key Points

The research aims to classify nonlinear evolution equations that allow quasi-local symmetries through an algebraic framework.
Developed an algebraic approach for classifying nonlinear evolution equations in one spatial dimension.
Analyzed transformation groups that involve integrals of the dependent variable.
Constructed inequivalent realizations of Lie algebras in two and three dimensions.
Identified nonlinear evolution equations admitting quasi-local symmetries.
Generalized the classification method for systems of evolution equations with two independent variables.

Abstract

We develop an efficient algebraic approach to classifying nonlinear evolution equations in one spatial dimension that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable.It applies to evolution equations invariant under Lie point symmetries leaving the temporal variable invariant.We construct inequivalent realizations of two-and three-dimensional Lie algebras leading to evolution equations admitting quasi-local symmetries.Finally, we generalize the approach in question for the case of an arbitrary system of evolution equations in two independent variables.

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