Geometrical formulation for adjoint-symmetries of partial differential equations

TL;DR

This paper introduces a geometric framework for understanding adjoint-symmetries of PDEs as 1-forms, revealing their duality with symmetries and their invariance properties in evolution systems, applicable in physics and mathematics.

Contribution

It develops a novel geometric formulation of adjoint-symmetries as 1-forms and explores their invariance properties in evolution equations with constraints.

Findings

Adjoint-symmetries are represented as 1-forms with geometric meaning.

Adjoint-symmetries in evolution systems are invariant under the flow.

The formulation applies to PDE systems in applied mathematics and physics.

Abstract

A geometrical formulation for adjoint-symmetries as 1-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by 1-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry 1-forms are shown to be invariant up to a functional multiplier of a normal 1-form associated to the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.