Hyperbolic differential-difference equations with nonlocal potentials

Abstract

We consider a three-parametric set of solutions for a two-dimensional hyperbolic differential-difference equation in a half-plane containing the sum of a differential operator and shift operators with respect to a spatial variable ranging on the entire real axis (or a differential-difference equation with nonlocal potentials). All shifts in potentials with respect to the spatial variable are arbitrary real numbers and no commensurability is assumed. This is the most general case. Nowadays, elliptic and parabolic functional-differential equations, and, in particular, differential-difference equations, are studied well enough. The aim of this work is to investigate hyperbolic differential-difference equations with shift operators in the space variable, which, as far as we know, have not been studied previously. The nature of the physical problems leading to such equations is fundamentally different from the problems for the classical equations of mathematical physics. To construct solutions, we employ a classical operation scheme is used, according to which the direct and then the inverse Fourier transforms are formally applied to the equation. However, if in the classical case the application of the Fourier transform leads to the study of polynomials with respect to the dual variable, in our case, due to the fact that in the Fourier images a shift operator is a multiplier, the symbol of the differential-difference operator is no longer a polynomial, but a combination of a power function and trigonometric functions with incommensurable arguments. This gives rise to computational difficulties and completely different effects in the solution. Generally speaking, this scheme leads to solutions in the sense of generalized functions. However, in this case it is possible to prove that the obtained solutions are classical. We prove a theorem that if the real part of the symbol of the differential-difference operator in the spatial variable involved in the equation is positive, then the constructed solutions are classical. Classes of equations for which this condition is satisfied are given. We obtain the relations for the coefficients and shifts in the equation ensuring the required positivity for the real part of the symbol of the differential-difference operator in the equation.