Higher Symmetries of the Wave Equation with Scalar and Vector Potentials

Higher order symmetry operators for the wave equation with scalar and vector potentials are investigated. All scalar potentials which admit second order symmetry operators are found explicitly. Symmetries of partial differential equations are used for separation of variables [1], description of conservation laws [2], construction of exact solutions [3], etc. Before can be applied, symmetries have to be found, therefore search for symmetries attracts attention of many investigators. We say a linear differential operator of order n is a symmetry operator (or simply a symmetry) of a partial differential equation if it transforms any solution of this equation into a solution. If n = 1, then the symmetry is nothing but a generator of a Lie group being a symmetry group of the equation considered. If n > 1, the related symmetries are referred as non-Lie or higher symmetries. The symmetry aspects of the Schrödinger equation have been investigated by many authors (see [1] and references cited in), the higher symmetries of this equation where investigated in [4–9]. In contrast, the higher symmetries of the relativistic wave equation have not been studied well yet. In this paper, we investigate the higher symmetries of the wave equation with an arbitrary scalar potential Lψ ≡ (∂μ∂ − V )ψ = 0 (1) where μ = 0, 1, . . . ,m. We deduce equations describing both potentials and coefficients of the corresponding symmetry operators of order n and present the complete list of potentials admitting symmetries for the case m = 1, n = 2, which completes the results of papers [10, 11]. In addition, we find all the possible scalar potentials V and vector potentials Aμ such that the equation L̂ψ ≡ (DμD − V )ψ = 0, Dμ = ∂μ − eAμ, μ = 0, 1 (2) 322 R. Andrushkiw and A. Nikitin admits any Lie symmetry. Moreover, we consider the case of time-dependent potential V and present a constructive test in order to answer the question if the corresponding wave equation admits any Lie symmetry. Let us represent a differential operator of arbitrary order n in the form