Properties of the solitonic potentials of the heat operator

Properties of the pure solitonic $\tau$-function and potential of the heat equation are studied in detail. We describe the asymptotic behavior of the potential and identify the ray structure of this asymptotic behavior on the $x$-plane in dependence on the parameters of the potential

B\"{a}cklund and Darboux transformations for the nonstationary Schr\"{o}dinger equation

Potentials of the nonstationary Schr\"{o}dinger operator constructed by means of $n$ recursive B\"{a}cklund transformations are studied in detail. Corresponding Darboux transformations of the Jost solutions are introduced. We show that these solutions obey modified integral equations and present their analyticity properties. Generated transformations of the spectral data are derived.Comment: to be published in Proc. of the Steklov Inst. of Mathematics, Moscow, Russi