QUASICLASSICAL LIMIT OF THE SCHRÖDINGER-MAXWELL- BLOCH EQUATIONS

The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

What is integrability of discrete variational systems?

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z -invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d -dimensional pluri-Lagrangian problem can be described as follows: given a d -form on an m -dimensional space (called multi-time, m > d ), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals for any d -dimensional manifold Σ in the multi-time. We derive the main building blocks of the multi-time Euler–Lagrange equations for a discrete pluri-Lagrangian problem with d =2, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. We analyse the system of corner equations for a special class of three-point two-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding two-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multi-dimensionally consistent system of quad-equations.

THE AHARONOV-BOHM EFFECT IN SCATTERING OF QUASICLASSICAL PARTICLES

Quantum-mechanical scattering of nonrelativistic charged particles by a magnetic vortex of nonzero transverse size is considered. The high-frequency limit of a scattered particle corresponds to the quasiclassical limit, and we show that the scattering Aharonov-Bohm effect persists in the quasiclassical limit owing to the Fraunhofer diffraction in the forward direction. Therefore, the flux of a magnetic vortex serves as a gate for the propagation of quasiclassical particles moving orthogonally to the vortex. The issue of the experimental detection of the Fraunhofer diffraction peak and the scattering Aharonov-Bohm effect is discussed.

BETHE ANSATZ FOR NON-COMPACT GROUPS: THE PAIRING MODELS FOR BOSONS

We discuss the construction of the exactly solvable pairing models for bosons in the framework of the Quantum Inverse Scattering method. It is stressed that this class of models naturally appears in the quasiclassical limit of the algebraic Bethe ansatz transfer matrix. We propose the new pairing Hamiltonians for bosons, depending on the additional parameters. It is pointed out that the new class of the pairing models can be obtained from the fundamental transfer-matrix. The possible new application of the pairing models for confined bosons in the physics of helium nanodroplets is pointed out.

CREATION OF SCALAR AND DIRAC PARTICLES IN THE PRESENCE OF A TIME VARYING ELECTRIC FIELD IN AN ANISOTROPIC UNIVERSE

We compute the density of scalar and Dirac particles created by a cosmological anisotropic universe1,2 in the presence of a time dependent homogeneous electric field. In order to compute the rate of particles created we apply a quasiclassical approach that has been used successfully in different scenarios3,4. The idea behind the method is the following: First, we solve the relativistic Hamilton-Jacobi equation and, looking at its solutions, we identify positive and negative frequency modes. Second, after separating variables5,6, we solve the Klein-Gordon and Dirac equations and, after comparing with the results obtained for the quasiclassical limit, we identify the positive and negative frequency states. We show that the particle distribution becomes thermal when one neglects the electric interaction.

STOCHASTIC EVOLUTION AS A QUASICLASSICAL LIMIT OF A BOUNDARY VALUE PROBLEM FOR SCHRÖDINGER EQUATIONS

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.