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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.
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We prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. This statement provides a significant generalization of results of the recent paper by Bluman, Cheviakov and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present extensions to gauged potential systems, Abelian and general coverings and general foliated systems of differential equations. An example illustrating possible applications of proved statements is considered. A special version of the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws.
We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral. They deform the spectral curve but preserve the multipliers of the Floquet functions. The latter property implies that the conservation laws, for soliton equations, which may be described in terms of the Floquet multipliers give rise to conservation laws for the corresponding equations with self-consistent sources. Such a property was first observed by us for some geometrical flow which appears in the conformal geometry of tori in three- and four-dimensional Euclidean spaces (math/0611215).
We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several discrete field theories: electrodynamics, gauge theory, Klein-Gordon and Dirac ones. In particular, we construct a conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains.
The 1 + 2 dimensional Bogoyavlensky-Konopelchenko Equation is investigated for its solution and conservation laws using the Lie point symmetry analysis. In the recent past, certain work has been done describing the Lie point symmetries for the equation and this work seems to be incomplete (Ray S (2017) Compt. Math. Appl. 74, 1157). We obtained certain new symmetries and corresponding conservation laws. The travelling-wave solution and some other similarity solutions are studied.
A systematic and unified approach to transformations and symmetries of general second order linear parabolic partial differential equations is presented. Equivalence group is used to derive the Appell type transformations, specifically Mehlers kernel in any dimension. The complete symmetry group classification is re-performed. A new criterion which is necessary and sufficient for reduction to the standard heat equation by point transformations is established. A similar criterion is also valid for the equations to have a four- or six-dimensional symmetry group (nontrivial symmetry groups). In this situation, the basis elements are listed in terms of coefficients. A number of illustrative examples are given. In particular, some applications from the recent literature are re-examined in our new approach. Applications include a comparative discussion of heat kernels based on group-invariant solutions and the idea of connecting Lie symmetries and classical integral transforms introduced by Craddock and his coworkers. Multidimensional parabolic PDEs of heat and Schrodinger type are also considered.
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