Exact Results in Quantum Theory

The Korteweg--de Vries (KdV) equation is a non-linear dispersive equation describing shallow-water waves and possessing many intriguing properties. One of them is existence of the so-called soliton solutions representing solitary waves travelling with constant speed and shape, as well as a special way in which several such solitons interact. Another interesting fact is that solutions of the KdV can be obtained as solutions of the inverse scattering problem for the family of associated Schroedinger operators, as discovered by S.Gardner, J.Green, M.Kruskal and R.Miura in 1967, and the classical soliton solutions of the KdV correspond precisely to the so-called reflectionless potentials (I.Kay and H.Moses, 1956).
The aim of this talk is two-fold. Firstly, we characterise the family of all Schroedinger operators with integrable reflectionless potentials and give an explicit formula producing all such potentials. Secondly, we use the inverse scattering transform approach to describe all solutions of the KdV equation whose initial (t=0) profile is an integrable reflectionless potential. Such solutions will stay integrable and reflectionless for all positive times and can be called generalized soliton solutions of KdV. This research extends and specifies in several ways the previous work on reflectionless potentials by V.Marchenko, C.Remling et al. and generalized soliton solutions of the KdV equation introduced by V.Marchenko in 1991 and F.Gesztesy, W.Karwowski, and Z.Zhao in 1992. The talk is based on a joint project with B.Melnyk and Ya.Mykytyuk (Lviv Franko National University).