Scattering by a potential using hyperbolic methods

Authors:
Alvin Bayliss, Yan Yan Li and Cathleen Synge Morawetz

Journal:
Math. Comp. **52** (1989), 321-338

MSC:
Primary 65M99; Secondary 65N99, 81F99

DOI:
https://doi.org/10.1090/S0025-5718-1989-0958869-1

MathSciNet review:
958869

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This is a preliminary note on a numerical time-dependent approach to the inverse scattering problem where the data given is back scattering. The results in one space variable are both computed and are quite comparable to the best existing results. A corresponding algorithm is developed for two variables. A major stage is the computation of the forward problem for the wave equation with potential and with an incoming plane wave. Interesting developments are a boundary layer next to the propagating delta function and effective resolution of the scattered wave. It is assumed there are no bound states.

**[1]**K. B. Bube & R. Burridge, "The one-dimensional inverse problem of reflection seismology,"*SIAM Rev.*, v. 25, 1983, pp. 497-559. MR**788323 (87c:86005)****[2]**R. Burridge, "The Gel'fand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems,"*Wave Motion*, v. 2, 1980, pp. 305-323. MR**593133 (82d:81144)****[3]**P. Deift & E. Trubowitz, "Inverse scattering on the line",*Comm. Pure Appl. Math.*, v. 32, 1979, pp. 121-251. MR**512420 (80e:34011)****[4]**L. D. Fadeev, "Inverse problems of quantum scattering theory. II,"*J. Soviet Math.*, v. 5, 1976, pp. 334-396.**[5]**Roger G. Newton, "Inverse scattering, Three dimensions II and III,"*J. Math. Phys.*, v. 21, 1980, pp. 1698-1715; v. 22, 1981, pp. 2191-2200. Also Erratum, v. 22, 1981, p. 631.**[6]**G. A. Kriegsmann & C. S. Morawetz, "The calculations of an inverse potential problems,"*SIAM J. Appl. Math.*, v. 43, 1983, pp. 844-854. MR**709741 (84h:35165)****[7]**R. T. Prosser, "Formal solutions of inverse scattering problems. I-IV,"*J. Math. Phys.*, v. 10, 1969, pp. 1819-1822; v. 17, 1976, pp. 1775-1779; v. 21, 1980, pp. 2648-2653; v. 23, 1982, pp. 2127-2130. MR**0503220 (58:20021)****[8]**D. C. Stickler, "Inverse scattering in a stratified medium,"*J. Acoust. Soc. Amer.*, v. 74, 1983, pp. 994-1005. MR**717863 (84k:76098)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M99,
65N99,
81F99

Retrieve articles in all journals with MSC: 65M99, 65N99, 81F99

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0958869-1

Keywords:
Scattering from a potential,
inverse problems

Article copyright:
© Copyright 1989
American Mathematical Society