Regularization of a Loaded Inverse Problem of the Korteweg-deVries Type in an Unbounded Domain

Abstract

This paper considers an inverse problem for a loaded differential operator of the Korteweg-de Vries type in an unbounded domain. It is shown that the problem reduces to a degenerate ill-posed nonlinear Volterra integral equation of the first kind of a special form.
The study is carried out using the Fourier transform and a regularization method in specially introduced functional spaces. The regularizability of the inverse problem is investigated in a generalized sense.
It is noted that such loaded inverse problems of the Korteweg-de Vries type, leading to ill-posed Volterra integral equations of the first kind, arise in wave theory, in particular in the study of propagation processes of dispersive loaded waves and related phenomena.