Inverse scattering problem for a hyperbolic system of equations on the semiaxis

• Дата: 2026
• Автори: Nizhnik, L., Iskenderov, N. Sh., Нижник, Леонід Павлович, Нижник, Леонід, Іскендеров, Нізамеддін Ш.
• Формат: Стаття
• Мова: Українська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2026
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/9744
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.98, 517.95 We study а scattering problem for the hyperbolic systems of first-order differential equations on the semiaxis under the assumption that all solutions exhibit asymptotic behaviors at infinity corresponding either to incident or to scattered waves. The problem is reduced to finding the solution of the system in the case where all incident waves are known together with a certain homogeneous boundary condition. Under these assumptions, the solution exists and is unique. Therefore, we observe the appearance of a scattering operator on the semiaxis, which transforms all incident waves into scattered waves. For a special case in which the numbers of incident and scattered waves are equal, we establish the detailed properties of the scattering operator, including its factorization into Volterra factors of the opposite polarities. This factorization makes it possible to solve the inverse scattering problem on the semiaxis, i.e., to unambiguously reconstruct all potentials in the system according to the two known scattering operators corresponding to two different boundary conditions.