Direct and inverse scattering on the line.

*(English)*Zbl 0679.34018
Mathematical Surveys and Monographs, 28. Providence, RI: American Mathematical Society (AMS). xiii, 209 p. (1988).

The goal of the book is to solve the inverse scattering problem for the nth order differential operator \(L_ n\) on the line. This problem is well understood from different points of view for \(n=2\). However, a generalization to arbitrary n meets with numerous difficulties. In particular, even the setting up of the problem needs to be made precise. In part I the authors study the forward problem when coefficients of the operator \(L_ n\) are known and its eigenfunctions should be constructed. The scattering data \(S(L_ n)\) arise as certain relations between eigenfunctions of \(L_ n\). In contrast to the special case \(n=2\) the consideration of unbounded eigenfunctions is required. Moreover, in the general case the study of the low-energy limit becomes essentially more complicated.

In part II the inverse problem is treated. Here two different questions should be distinguished. The first is to characterize scattering data corresponding to differential operators. The second is to give a procedure for reconstruction of coefficients of \(L_ n\) when \(S(L_ n)\) is known. Both these problems are considered in the book. It twins out that the cases of even n(n\(\geq 4)\) and odd n should be studied separately. From analytic view point the inverse problem is similar to a matrix factorization Riemann-Hilbert problem. More precisely, it is treated as the so-called \({\bar \partial}\)-problem. Special attention is paid to the case of self-adjoint \(L_ n\). The final part III is devoted to applications of the results obtained. The most important of them is an analysis of nonlinear equations of KdV type.

In part II the inverse problem is treated. Here two different questions should be distinguished. The first is to characterize scattering data corresponding to differential operators. The second is to give a procedure for reconstruction of coefficients of \(L_ n\) when \(S(L_ n)\) is known. Both these problems are considered in the book. It twins out that the cases of even n(n\(\geq 4)\) and odd n should be studied separately. From analytic view point the inverse problem is similar to a matrix factorization Riemann-Hilbert problem. More precisely, it is treated as the so-called \({\bar \partial}\)-problem. Special attention is paid to the case of self-adjoint \(L_ n\). The final part III is devoted to applications of the results obtained. The most important of them is an analysis of nonlinear equations of KdV type.

Reviewer: D.Yafaev