Abstract:
We construct a three-parameter family of smooth solutions for a two-dimensional hyperbolic differential-difference equation considered in a half-plane and containing the sum of a differential operator and shift operators with respect to a spatial variable varying on the entire real axis.
We use a classical operating scheme, 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, then in this case, 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 led 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 the theorem ithat if the real part of the symbol of the differential-difference operator is positive, then the constructed solutions are classical. We dive classes of equations for which this condition is satisfied.
