Symmetries of the Hirota Difference Equation

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as "times" of the nonlinear integrable partial differential-difference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented

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