Conservation Laws and Potential Symmetries of Linear Parabolic Equations

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.

arXiv comment67 pages, minor corrections

References in corpus (8)

• Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations
• New results on group classification of nonlinear diffusion-convection equations
• Hierarchy of Conservation Laws of Diffusion--Convection Equations
• Potential equivalence transformations for nonlinear diffusion-convection equations
• Connections Between Symmetries and Conservation Laws
• The structure of the category of parabolic equations
• The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations
• Equivalence of conservation laws and equivalence of potential systems

Cited by in corpus (40)

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• Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations
• More common errors in finding exact solutions of nonlinear differential equations. I
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• Conservation laws and normal forms of evolution equations
• Group classification of linear evolution equations
• Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusion
• Reduction Operators of Linear Second-Order Parabolic Equations
• Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations
• Extended symmetry analysis of generalized Burgers equations
• Point and generalized symmetries of the heat equation revisited
• Enhanced symmetry analysis of two-dimensional Burgers system
• Local conservation laws of second-order evolution equations
• Point symmetry group of the barotropic vorticity equation
• Group classification of variable coefficient generalized Kawahara equations
• Extended symmetry analysis of two-dimensional degenerate Burgers equation
• Extended symmetry analysis of isothermal no-slip drift flux model
• Inverse problem on conservation laws
• Potential Conservation Laws
• Equivalence groupoids of classes of linear ordinary differential equations and their group classification
• Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
• Conservation laws and hierarchies of potential symmetries for certain diffusion equations
• Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
• Group Analysis of Variable Coefficient Diffusion--Convection Equations. III. Conservation Laws
• Conservative parameterization schemes
• Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model
• Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations
• Mapping method of group classification
• Group Analysis of Variable Coefficient Diffusion-Convection Equations. IV. Potential Symmetries
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
• Normalized classes of generalized Burgers equations
• Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation
• Extended symmetry analysis of a "nonconservative Fokker-Plank equation"
• Equivalence and Symmetries for Linear Parabolic Equations and Applications Revisited
• Parameter-dependent linear ordinary differential equations and topology of domains
• Simplest potential conservation laws of linear evolution equations
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity
• Generalized Baecklund-Darboux transformation: conservation laws, rational extensions and bispectrality