Inverse scattering problem for the Schroedinger equation on the axis

Necessary and sufficient conditions on the reflection coefficient are obtained for the Schroedinger equation on the whole axis with a potential growing on one side by utilization of the inverse problem formalism.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 677–684, December, 1968.The author is grateful to L. D. Faddeev for formulating the problem, and for constant attention, and to V. S. Buslaev, whose remarks have been utilized.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.     

Inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition

We give a complete solution of the inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition in the absence of discrete spectrum.     

Inverse viscosity problem for the Navier-Stokes equation

We consider an inverse problem of determining a viscosity coefficient in the Navier-Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.     

Inverse problem for a parabolic equation

Mathematical Notes -     

Inverse problem for the heat equation with degeneration

We consider the inverse problem of determining the time-dependent thermal diffusivity that is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1563–1570, November, 2005.     

Inverse scattering problem for nonstationary Dirac-type systems on the half-plane

In this paper was considered the scattering problem for the nonstationary Dirac-type systems of n (n?2) equations on the half-plane when the system has n₁ (1?n₁?n−1) incident and n₂ (n₂=n−n₁) scattered waves. In case n₁ is divisible by n₂, we formulate the inverse scattering problem for a nonstationary Dirac-type system when considering m () scattering problems on the half-plane with the same incident waves but different boundary conditions. Moreover, the scattering operator for the nonstationary Dirac-type system on half-plane was defined and unique restoration of the potential with respect to the scattering operator was proved.     

Inverse scattering problem for nonstationary Dirac-type systems on the plane

In this paper the forward and inverse scattering problems for the nonstationary Dirac-type systems on the plane are considered. The scattering data for the inverse scattering problem (ISP) is defined and a unique restoration of the potential from the scattering data is proved.