Abstract:
We investigate a hyperbolic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. Such differential-difference equations arise in various applications not covered by the classical theory of differential equations. On the other hand, they are quite interesting from the theoretical viewpoint because of specific effects caused by the nonlocal nature of the investigated equations. We find a condition for the vector of coefficients at nonlocal terms of the investigated equation and the translation vectors, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation; two of the specified parameters are real values, while the remaining one is a real-coordinate vector such that its dimension is equal to the amount of nonlocal terms (i. e., translated potentials) of the investigated equation. No commensurability requirements are imposed on the coefficients at nonlocal terms of the equation. Acknowledgements The work is supported by the Ministry of Science and Higher Education of the Russian Federation (project number FSSF-2023-0016).
