On Hyperbolic Equations with Arbitrarily Directed Translations of Potentials

Abstract

We study a hyperbolic equation with an arbitrary number of potentials undergoing translation in arbitrary directions. Differential-difference equations arise in various applications that are not covered by the classical theory of differential equations. In addition, they are of considerable interest from a theoretical point of view, since the nonlocal nature of such equations gives rise to various effects that do not arise in the classical case. We find a condition on the vector of coefficients for nonlocal terms in the equation and on the vectors of potential translations that ensures the global solvability of the equation under consideration. By imposing the specified condition on the equation and using the classical Gelfand–Shilov scheme, we explicitly construct a three-parameter family of smooth global solutions to the equation under study.