Conservation laws, potential symmetries and Darboux transformation of linear parabolic equations

Abstract:
Extended investigation on conservation laws and potential symmetries is performed for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised with usage of 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 exhausted by local ones. For any equation under consideration the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria on existence of potential symmetries are proposed. Their proofs involves a sophisticated technique based on 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. 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.

Joint work with Michael Kunzinger (University of Vienna, Austria) and Nataliya Ivanova (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine).