Generalization of Sato equation and systems of multidimensional nonlinear Partial Differential Equations

This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE) together with solutions parameterized by the set of arbitrary functions.Comment: 10 page

Informational correlation between two parties of a quantum system: short spin-1/2 chains with XY Hamiltonian

We introduce the informational correlation $E^{AB}$ between two interacting quantum subsystems $A$ and $B$ of a quantum system as the number of arbitrary parameters $\varphi_i$ of a unitary transformation $U^A$ (locally performed on the subsystem $A$) which may be detected in the subsystem $B$ by the local measurements. This quantity indicates whether the state of the subsystem $B$ may be effected by means of the unitary transformation applied to the subsystem $A$. Emphasize that $E^{AB}\neq E^{BA}$ in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem $B$. However, there is some non-reducible part of $E^{AB}(t)$ which may not be decreased by any unitary transformation of the subsystem $B$ at a fixed time instant $t$. Two examples of the informational correlations between two parties of the four node spin-1/2 chain are studied.Comment: 45 pages, 2 figure

On integration of multidimensional generalizations of classical CC- and SS-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.Comment: 31 page

Partial structural restoring of two-qubit transferred state

We consider the communication line with two-qubit sender and receiver, the later is embedded into the four-qubit extended receiver. Using the optimizing unitary transformation on the extended receiver we restore the structure of the non-diagonal part of an arbitrary initial sender's state at the remote receiver at certain time instant. Obstacles for restoring the diagonal part are discussed. We represent examples of such structural restoring in a communication line of 42 spin-1/2 particles.Comment: 17 pages, 2 figure

A five-dimensional solitary-wave first order nonlinear PDE integrable by dressing method

We derive a five-dimensional nonlinear first order matrix PDE which is a generalization of the completely integrable (2+1)-dimensional $N$-wave equation. Similar to the $\bar\partial$-problem, our algorithm is based on the linear integral equation of special form.Comment: 7 page

Matrix equations of hydrodynamic type as lower-dimensional reductions of Self-dual type SS-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.Comment: 44 page

Coherence evolution and transfer supplemented by state-restoring

The evolution of quantum coherences comes with a set of conservation laws provided that the Hamiltonian governing this evolution conserves the spin-excitation number. At that, coherences do not intertwist during the evolution. Using the transmission line and the receiver in the initial ground state we can transfer the coherences to the receiver without interaction between them, { although the matrix elements contributing to each particular coherence intertwist in the receiver's state. } Therefore we propose a tool based on the unitary transformation at the receiver side to { untwist these elements and thus} restore (at least partially) the structure of the sender's initial density matrix. A communication line with two-qubit sender and receiver is considered as an example of implementation of this technique.Comment: 18 page

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.Comment: 14 page

Construction of particular solutions to nonlinear equations of Mathematical Physics by using matrix algebraic equation

The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between algebraic matrix equations and PDE. Example of (2+2)-dimensional generalization of Burgers equation is given.Comment: 17 page

Remote control of quantum correlations in two-qubit receiver via three-qubit sender

We study the problem of remote control of quantum correlations (discord) in a sub-system of two qubits (receiver) via the parameters of the initial state of another sub-system of three qubits (sender) connected with the receiver by the inhomogeneous spin-1/2 chain. We propose two parameters characterizing the creatable correlations. The first one is the discord between the receiver and the rest of spin-1/2 chain, it concerns the mutual correlations between these two subsystems. The second parameter is the discord between the two nodes of the receiver and describes the correlations inside of the receiver. We study the dependence of these two discords on the inhomogeneity degree of spin chain.Comment: 19 pages, 6 figure