The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of Novikov in Int. Math. Res. Papers 2005(6):287–349, 2005]] and implemented numerically in Alekseenko et al. in Acoust. J. 54(3), 2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, 1985]] and Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, 1987]] and involves, in particular, some results of Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, 1965], 1974]] and some ideas of the soliton theory (in particular, some ideas going back to Manakov in Usp. Mat. Nauk 31(5):245–246, 1976]] and Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, 1976]]). Also, our studies go back, in particular, to Regge in Nuovo Cimento 14:951–976, 1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.