Conservation Laws and Potential Symmetries of Linear Parabolic Equations

TL;DR

This paper thoroughly investigates conservation laws and potential symmetries of linear (1+1)-dimensional second-order parabolic equations, classifying potential symmetries and establishing criteria for their existence.

Contribution

It provides a complete description of potential conservation laws, revises group classification using advanced methods, and analyzes potential symmetries for a broad class of equations.

Findings

All potential conservation laws are local and exhaustively described.

Criteria for potential symmetry existence are established.

Potential symmetries of the linear heat equation are classified.

Abstract

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.