arXiv subjects
Publications and source records attributed to A. K. Pogrebkov.
In \textit{SIGMA} \textbf{17} (2021), 091, 12 p.p.\ we have presented an integrable system with a negative time variable number for the Davey-Stewartson hierarchy. Here we develop this approach to construct an integrable equation with a lower time variable number. In addition, we show that the system reduced by this time is a new integrable equation in the dimension $1+1$.
Induced dynamics is defined as dynamics of real zeros with respect to $x$ of equation $f(q_1-x,\ldots,q_N-x,p_1,\ldots,p_N)=0$, where $f$ is a function, and $q_i$ and $p_j$ are canonical variables obeying some (free) evolution. Identifying zero level lines with the world lines of particles, we show that the resulting dynamical system demonstrates highly nontrivial collisions of particles. In particular, induced dynamical systems can describe such ``quantum'' effects as bound states and creation/annihilation of particles, both in nonrelativistic and relativistic cases. On the other side, induced dynamical systems inherit properties of the $(p,q)$-systems being Hamiltonian and Liouville integrable.
Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function in the case of pure solitonic solution is given and properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as example. This enables formulation of the Darboux transformations in terms of the Cauchy-Jost function and classification of these transformations. Action of Darboux transformations on Grassmanians-i.e., on the space of soliton parameters-is derived and relation of the Darboux transformations with property of total nonnegativity of elements of corresponding Grassmanians is discussed.
Properties of the Cauchy--Jost (known also as Cauchy--Baker--Akhiezer) function of the KPII equation are described. By means of the $\bar\partial$-problem for this function it is shown that all equations of the KPII hierarchy are given in a compact and explicit form, including equations on the Cauchy--Jost function itself, time evolutions of the Jost solutions and evolutions of the potential of the heat equation.
The Direct and the Inverse Scattering Problems for the heat operator with a potential being a perturbation of an arbitrary $N$ soliton potential are formulated. We introduce Jost solutions and spectral data and present their properties. Then, giving the time evolution of the spectral data, the initial value problem of the Kadomtsev-Petviashvili II equation for a solution describing $N$ solitons perturbed by a generic smooth fast decaying potential is linearized.
The heat operator with a general multisoliton potential is considered and its extended resolvent, depending on a parameter $q\in\R^2$ is derived. Its boundedness properties in all variables and its discontinuities in the parameter $q$ are given. As the result, the Green's functions are introduced and their properties are studied in detail.
The heat operator with a pure soliton potential is considered and its Green's function, depending on a complex spectral parameter k, is derived. Its boundedness properties in all variables and its singularities in the spectral parameter k are studied. A generalization of the Green's function, the extended resolvent, is also given.
Properties of the pure solitonic $τ$-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.
Twisting transformations for the heat operator are introduced. They are used, at the same time, to superimpose a` la Darboux N solitons to a generic smooth, decaying at infinity, potential and to generate the corresponding Jost solutions. These twisting operators are also used to study the existence of the related extended resolvent. Existence and uniqueness of the extended resolvent in the case of $N$ solitons with N "ingoing" rays and one "outgoing" ray is studied in details.
Developing observation made in \cite{commut} we show that simple identity of the commutator type on an associative algebra is in one-to-one correspondence to 2D (infinite) Toda chain. We introduce representation of elements of associative algebra that, under some generic conditions, enables derivation of the Toda chain equation and its Lax pair from the given commutator identity.
It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to associate to such commutator identity both nonlinear equation and its Lax pair. Thus problem of construction of new integrable pde's reduces to construction of commutator identities on associative algebras.
In the framework of the resolvent approach it is introduced a so called twisting operator that is able, at the same time, to superimpose à la Darboux $N$ solitons to a generic smooth decaying potential of the Nonstationary Schrödinger operator and to generate the corresponding Jost solutions. This twisting operator is also used to construct an explicit bilinear representation in terms of the Jost solutions of the related extended resolvent. The main properties of the Jost and auxiliary Jost solutions and of the resolvent are discussed.
The Nonstationary Schrödinger equation with potential being a perturbation of a generic one-dimensional potential by means of a decaying two-dimensional function is considered here in the framework of the extended resolvent approach. The properties of the Jost solutions and spectral data are investigated.
Inverse scattering transform method of the heat equation is developed for a special subclass of potentials nondecaying at space infinity---perturbations of the one-soliton potential by means of decaying two-dimensional functions. Extended resolvent, Green's functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the Inverse problem is formulated in an exact distributional sense.
The set of integrable symmetries of the nonstationary Schrödinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets ultimately lead to nonlinear (possibly, operator) equations of the Kadomtsev--Petviashvili I type or its higher analogues, thus demonstrating that the linear problem itself constructively determines the associated nonlinear integrable evolution equations and their hierarchies.
Ablowitz-Ladik linear system with range of potential equal to {0,1} is considered. The extended resolvent operator of this system is constructed and the singularities of this operator are analyzed in detail.
The discrete Schrödinger equation with potential belonging to $\F_{2}$ is solved explicitly. On this base the associated (1+1)-dimensional cellular automaton is examined and corresponding set of integrals of motions is constructed.