arXiv subjects

A. Pogrebkov

Publications and source records attributed to A. Pogrebkov.

5 recordsLinked to original sources

arXiv2011

Heat operator with pure soliton potential: properties of Jost and dual Jost solutions

Properties of Jost and dual Jost solutions of the heat equation, $Φ(x,k)$ and $Ψ(x,k)$, in the case of a pure solitonic potential are studied in detail. We describe their analytical properties on the spectral parameter $k$ and their asymptotic behavior on the $x$-plane and we show that the values of $e^{-qx}Φ(x,k)$ and the residua of $e^{qx}Ψ(x,k)$ at special discrete values of $k$ are bounded functions of $x$ in a polygonal region of the $q$-plane. Correspondingly, we deduce that the extended version $L(q)$ of the heat operator with a pure solitonic potential has left and right annihilators for $q$ belonging to these polygonal regions.

nlin.SI↗

arXiv2002

Hierarchy of quantum explicitly solvable and integrable models

Realizing bosonic field v(x) as current of massless (chiral) fermions we derive hierarchy of quantum polynomial interactions of the field v(x) that are completely integrable and lead to linear evolutions for the fermionic field. It is proved that in the classical limit this hierarchy reduces to the dispersionless KdV hierarchy. Application of our construction to quantization of generic completely integrable interaction is demonstrated by example of the mKdV equation.

nlin.SI↗

arXiv2001

On quantization of the KdV equation

Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by means of the fermionic representation for the KdV field is considered. It is shown that in both cases the corresponding Hamiltonians are given as sums of two well defined operators. Each of them is bilinear and diagonal with respect to either fermion, or boson (current) creation-annihilation operators. As a result the quantization procedure needs no any space cut-off and can be performed on the whole axis. Existence of the solitonic states in the Hilbert space as well as quantization of soliton parameters are shown to result from this approach. As a by-product it is also demonstrated that the dispersionless KdV is uniquely and explicitly solvable in the quantum case.

nlin.SI↗

arXiv1999

Bäcklund and Darboux transformations for the nonstationary Schrödinger equation

Potentials of the nonstationary Schrödinger operator constructed by means of $n$ recursive Bäcklund 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.

math-ph↗

arXiv1993

Solutions of the Kpi Equation with Smooth Initial Data

The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with given initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No additional special constraints, usually considered in literature, as $\int\!dx\,u(0,x,y)=0$ are required to be satisfied by the initial data. The problem is completely solved in the framework of the spectral transform theory and it is shown that $u(t,x,y)$ satisfies a special evolution version of the KPI equation and that, in general, $\partial_t u(t,x,y)$ has different left and right limits at the initial time $t=0$. The conditions of the type $\int\!dx\,u(t,x,y)=0$, $\int\!dx\,xu_y(t,x,y)=0$ and so on (first, second, etc. `constraints') are dynamically generated by the evolution equation for $t\not=0$. On the other side $\int\!dx\!\!\int\!dy\,u(t,x,y)$ with prescribed order of integrations is not necessarily equal to zero and gives a nontrivial integral of motion.

solv-int↗