No abstract

The spectral transform for the nonstationary Schrodinger equation is considered. The resolvent operator of the Schrodinger equation is introduced and the Fourier transform of its kernel (called the resolvent function) is studied. It is shown that it can be used to construct a generalized version of the theory of the spectral transform which enables one to handle also potentials approaching zero in every direction except a finite number, which corresponds to the physical situation of long waves mutually interacting in the plane.

No abstract

A reduction of the Davey-Stewartson I1 equation to a (2 + 0)-dimensional Liouville equation is proposed. A class of singular solutions with singularities lying on the closed curves ('closed string-like') is given and discussed.

Bicklund transformations of the timedependent SchrBdinger equation which transform a real potential into another real potential are constructed, as well as their Darboux versions. The iterated application of these Bicklund transformations lo a generic potential is considered and the obtained recursion relations are explicitly solved. It is shown that the dressing of the generic potential can be obtained by taking the continuous limit of this infinite sequence of Bicklund transformations, and that the J-bar integral equations, which solve the inverse spectral problem, can be obtained as thc continuous limit of the recurrence equation defining the sequence of the Darboux transformations.