The one-dimensional inverse scattering problem for nonhomogeneous media with discontinuous wave speed. (English) Zbl 0883.34077

Aktosun, Tuncay; Klaus, Martin; van der Mee, Cornelis

Hinton, Don (ed.) et al., Spectral theory and computational methods of Sturm-Liouville problems. Proceedings of the 1996 conference, Knoxville, TN, USA, in conjunction with the 26th Barrett memorial lecture series. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 191, 105-133 (1997).

Summary: We consider the inverse problem of reconstructing the wave speed in a one-dimensional nonhomogeneous medium from appropriate scattering data. The wave speed is allowed to have jump discontinuities and the medium may be subject to a nonhomogeneous external restoring force. In the frequency-domain this inverse problem leads to a Riemann-Hilbert problem and an associated singular integral equation. Under suitable conditions we prove that the singular integral equation is uniquely solvable and we discuss how its solution leads to the recovery of the wave speed. We also show that certain characteristic properties of the wave speed can be reconstructed more quickly, that is, without completely solving the inverse problem first. Some examples illustrating the reconstruction of the wave speed are presented.
For the entire collection see [Zbl 0866.00046].

MSC:

34L25	Scattering theory, inverse scattering involving ordinary differential operators
81U40	Inverse scattering problems in quantum theory
34A55	Inverse problems involving ordinary differential equations
78A45	Diffraction, scattering
35Q15	Riemann-Hilbert problems in context of PDEs
45E99	Singular integral equations

Keywords:

inverse problem; one-dimensional nonhomogeneous medium; scattering data; jump discontinuities; Riemann-Hilbert problem; associated singular integral equation