Conservation Laws and Potential Symmetries of Linear Parabolic Equations
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Conservation Laws and Potential Symmetries of Linear Parabolic Equations Roman O. Popovych · Michael Kunzinger · Nataliya M. Ivanova
Received: 30 August 2007 / Accepted: 9 November 2007 / Published online: 27 November 2007 © Springer Science+Business Media B.V. 2007
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. Keywords Lie symmetry of differential equations · Group classification problem · Potential symmetry of differential equations · Conservation laws · Fokker–Planck equations · Kolmogorov equations · Darboux transformation · Equivalence group · Admissible transformation · Normalized class of differential equations · Generalized potential symmetry of differential equations R.O. Popovych () · N.M. Ivanova Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kiev 4, Ukraine e-mail: [email protected] N.M. Ivanova e-mail: [email protected] R.O. Popovych · M. Kunzinger Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austria M. Kunzinger e-mail: [email protected]
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1 Introduction In the present paper we classify local and potential conservation laws and potential symmetries of linear (1 + 1)-dimensional homogeneous second-order parabolic equations of the general form ut = A(t, x)uxx + B(t, x)ux + C(t, x)u,
(1)
where A = A(t, x), B = B(t, x) and C = C(t, x) are arbitrary smooth functions, A = 0. This class contains a number of physically important subclasses that are widely investigated and may be applied in many situations. Probably the most famous examples are the Kolmogorov equations (C = 0) and adjoint to them the Fokker–Planck equations (Axx
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