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A. I. Zenchuk

Publications and source records attributed to A. I. Zenchuk.

At least 73 records · Page 4Linked to original sources
arXiv2010

On integration of multidimensional generalizations of classical $C$- and $S$-integrable nonlinear partial differential equations

We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of (1+1)-dimensional $C$-integrable and (2+1)-dimensional $S$-integrable $N$-wave equations are derived among examples. Examples of multidimensional second order PDEs are represented as well.

nlin.SI↗
arXiv2010

The high probability state transfers and entanglements between different nodes of the homogeneous spin 1/2 chain in inhomogeneous external magnetic field

We consider the high probability state transfers and entanglements between different nodes of the spin-1/2 chains governed by the XXZ-Hamiltonian using the inhomogeneous stationary external magnetic field. Examples of three-, four-, ten- and twenty-node chains are represented. A variant of realization of the proper inhomogeneous magnetic field is discussed.

quant-ph↗
arXiv2009

Differential reductions of the Kadomtsev-Petviashvili equation and associated higher dimensional nonlinear PDEs

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with they scalar and three-dimensional reductions. Variants of the Kadomtsev-Petviashvili type and Korteweg-de Vries type equations are represented among them. Our algorithm is based on the combination of two Frobenius type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.

nlin.SI↗
arXiv2009

Multiple quantum NMR dynamics of spin-1/2 carrying molecules of a gas in nanopores

We consider the multiple quantum (MQ) NMR dynamics of a gas of spin carrying molecules in nanocavities. MQ NMR dynamics is determined by the residual dipole-dipole interactions which are not averaged completely due to the molecular diffusion in nanopores. Since the averaged non-secular Hamiltonian describing MQ NMR dynamics depends on only one coupling constant, this Hamiltonian commutes with the square of the total spin angular momentum $\hat I^2$. We use the basis of common eigenstates of $\hat I^2$ and the projection of $I$ on the external magnetic field for investigation of MQ NMR dynamics. This approach allows us to study MQ NMR dynamics in systems consisting of several hundreds of spins. The analytical approximation of the stationary profile of MQ coherences is obtained. The analytical expressions for MQ NMR coherence intensities of the five-spin system in a nanopore are found. Numerical investigations allow us to find the dependencies of intensities of MQ coherences on their orders (the profiles of MQ coherences) in systems consisting of 600 spins and even more. It is shown that the stationary MQ coherence profile in the considered system is an exponential one.

quant-ph↗
arXiv2009

On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields

It was shown recently that Frobenius reduction of the matrix fields reveals interesting relations among the nonlinear Partial Differential Equations (PDEs) integrable by the Inverse Spectral Transform Method ($S$-integrable PDEs), linearizable by the Hoph-Cole substitution ($C$-integrable PDEs) and integrable by the method of characteristics ($Ch$-integrable PDEs). However, only two classes of $S$-integrable PDEs have been involved: soliton equations like Korteweg-de Vries, Nonlinear Shrödinger, Kadomtsev-Petviashvili and Davey-Stewartson equations, and $GL(N,\CC)$ Self-dual type PDEs, like Yang-Mills equation. In this paper we consider the simple five-dimensional nonlinear PDE from another class of $S$-integrable PDEs, namely, scalar nonlinear PDE which is commutativity condition of the pair of vector fields. We show its origin from the (1+1)-dimensional hierarchy of $Ch$-integrable PDEs after certain composition of Frobenius type and differential reductions imposed on the matrix fields. Matrix generalization of the above scalar nonlinear PDE will be derived as well.

nlin.SI↗
arXiv2009

Relationship between probabilities of the state transfers and entanglements in spin systems with simple geometrical configurations

In this paper we derive analytical relations between probabilities of the excited state transfers and entanglements calculated by both the Wootters and positive partial transpose (PPT) criteria for the arbitrary spin system with single excited spin in the external magnetic field and Hamiltonian commuting with $I_z$. We apply these relations to study the arbitrary state transfers and entanglements in the simple systems of nuclear spins having two- and three-dimensional geometrical configurations with $XXZ$ Hamiltonian. It is shown that High-Probability State Transfers (HPSTs) are possible among all four nodes placed in the corners of the rectangle with the proper ratio of sides as well as among all eight nodes placed in the corners of the parallelepiped with the proper ratio of sides. Entanglements responsible for these HPSTs have been identified.

quant-ph↗
arXiv2008

On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation

We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf-Cole transformation. More concretely, 1) we show that the integrability properties (Lax pair, infinitely-many commuting symmetries, large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the Inverse Scattering Transform method ($S$-integrable) can be generated by the integrability properties of the (1+1)-dimensional matrix Bürgers hierarchy, integrable by the matrix Hopf-Cole transformation ($C$-integrable). 2) We show that the integrability properties i) of $S$-integrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of the $GL(M,\CC)$ self-dual Yang Mills equations, and iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced multidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius matrix, followed by a systematic elimination procedure of some of these blocks. The construction of large classes of solutions of the soliton equations from solutions of the matrix Bürgers hierarchy turns out to be intimately related to the construction of solutions in Sato theory. 3) We finally show that suitable generalizations of the block Frobenius matrix reduction of the matrix Bürgers hierarchy generates PDEs exhibiting integrability properties in common with both $S$- and $C$- integrable equations.

nlin.SI↗
arXiv2008

Matrix equations of hydrodynamic type as lower-dimensional reductions of Self-dual type $S$-integrable systems

We show that matrix $Q\times Q$ Self-dual type $S$-integrable Partial Differential Equations (PDEs) possess a family of lower-dimensional reductions represented by the matrix $ Q \times n_0 Q$ quasilinear first order PDEs solved in \cite{SZ1} by the method of characteristics. In turn, these PDEs admit two types of available particular solutions: (a) explicit solutions and (b) solutions described implicitly by a system of non-differential equations. The later solutions, in particular, exhibit the wave profile breaking. Only first type of solutions is available for (1+1)-dimensional nonlinear $S$-integrable PDEs. (1+1)-dimensional $N$-wave equation, (2+1)- and (3+1)-dimensional Pohlmeyer equations are represented as examples. We also represent a new version of the dressing method which supplies both classical solutions and solutions with wave profile breaking to the above $S$-integrable PDEs.

nlin.SI↗
arXiv2007

New multidimensional partially integrable generalization of S-integrable N-wave equation

This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. Method allows one to construct the systems of multidimensional Partial Differential Equations (PDEs) having the differential polynomial forms in any dimension n. Associated solution space is not full, although it is parametrized by a certain number of arbitrary functions of (n-1)-variables. We consider 4-dimensional generalization of the classical (2+1)-dimensional S-integrable N-wave equation as an example.

nlin.SI↗
arXiv2007

Dressing method based on homogeneous Fredholm equation: quasilinear PDEs in multidimensions

In this paper we develop a dressing method for constructing and solving some classes of matrix quasi-linear Partial Differential Equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In the simplest examples, the above dressing scheme captures matrix equations integrated by the characteristics method and nonlinear PDEs associated with matrix Hopf-Cole transformations.

nlin.SI↗
arXiv2006

The general solution of the matrix equation $w_t+\sum\limits_{k=1}^nw_{x_k}ρ^{(k)}(w)=ρ(w)+[w,T\tildeρ(w)]$

We construct the general solution of the equation $w_t+\sum\limits_{k=1}^nw_{x_k}ρ^{(k)}(w)=ρ(w)+[w,T\tildeρ(w)]$, for the $N\times N$ matrix $w$, where $T$ is any constant diagonal matrix, $n, N \in \NN_+$ and $ρ^{(k)}, ρ, \tildeρ: \RR \to \RR$ are arbitrary analytic functions. Such a solution is based on the observation that, as $w$ evolves according to the above equation, the evolution of its spectrum decouples, and it is ruled by the scalar analogue of the above equation. Therefore the eigenvalues of $w$ and suitably normalized eigenvectors are the $N^2$ Riemann invariants. We also obtain, in the case $ρ=\tildeρ=0$, a system of $N^2$ non-differential equations characterizing such a general solution. We finally discuss reductions of the above matrix equation to systems of $N$ equations admitting, as Riemann invariants, the eigenvalues of $w$. The simplest example of such reductions is a particular case of the gas dynamics equations

nlin.SI↗
arXiv2006

Partially integrable systems in multidimensions by a variant of the dressing method. 1

In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially integrable''. Such a construction is achieved using a suitable modification of the classical dressing scheme, consisting in assuming that the kernel of the basic integral operator of the dressing formalism be nontrivial. This new hypothesis leads to the construction of: 1) a linear system of compatible spectral problems for the solution $U(λ;x)$ of the integral equation in 3 independent variables each (while the usual dressing method generates spectral problems in 1 or 2 dimensions); 2) a system of nonlinear partial differential equations in $n$ dimensions ($n>3$), possessing a manifold of analytic solutions of dimension ($n-2$), which includes one largely arbitrary relation among the fields. These nonlinear equations can also contain an arbitrary forcing.

nlin.SI↗
arXiv2006

A variant of the Dressing Method applied to nonintegrable multidimensional nonlinear Partial Differential Equations

We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single linear Partial Differential Equation (PDE) with potentials expressed through the field of the nonlinear PDE. The absence of linear overdetermined system associated with nonlinear PDE creates an obstacle to obtain evolution of the spectral data (or dressing functions): evolution is defined by nonlinear IDE (or PDE in particular case). As an example, we consider generalization of the dressing method applicable to integrable (2+1)-dimensional $N$-wave and Davey-Stewartson equations. Although represented algorithm does not supply an analytic particular solutions, this approach may have a perspective development.

math.AP↗
arXiv2003

Generalized hierarchy of matrix Burgers type and $n$-wave equations

This article concerns the dressing method for solving of multidimensional nonlinear Partial Differential Equations. In particular, we join hierarchy of matrix Burgers type equation with hierarchies of equations integrable by the Inverse Spectral Transform (IST). Example of resonance interaction of wave packets in (3+1)-dimensions is given.

nlin.SI↗