We develop the method for constructing solutions to the nonlocal nonlinear Schrödinger equation (NLSE) with an anti-Hermitian term that are semiclassically localized on a one-dimensional manifold (a curve). The evolution of the curve is given by the closed system of integro-differential equations that can be treated as the “classical” analog of the open quantum system with the nontrivial geometry. Using our approach, we consider the evolution of vortex states in the open quantum system described by the specific model NLSE. The semiclassical stage of the vortex evolution can be treated as a quasi-steady vortex state. We show that the behavior of this state is largely determined by the geometry of the localization curve.

We deal with the n-dimensional nonlinear Schrödinger equation (NLSE) with a cubic nonlocal nonlinearity and an anti-Hermitian term, which is widely used model for the study of open quantum system. We construct asymptotic solutions to the Cauchy problem for such equation within the formalism of semiclassical approximation based on the Maslov complex germ method. Our solutions are localized in a neighbourhood of few points for every given time, i.e. form some spatial pattern. The localization points move over trajectories that are associated with the dynamics of semiclassical quasiparticles. The Cauchy problem for the original NLSE is reduced to the system of ODEs and auxiliary linear equations. The semiclassical nonlinear evolution operator is derived for the NLSE. The general formalism is applied to the specific one-dimensional NLSE with a periodic trap potential, dipole-dipole interaction, and phenomenological damping. It is shown that the long-range interactions in such model, which are considered through the interaction of quasiparticles in our approach, can lead to drastic changes in the behaviour of our asymptotic solutions.

For the population Fisher - Kolmogorov - Petrovsky - Piskunov equation with the nonlocal competitive losses and Caputo fractional time derivative of the order 0 < α < 1, dynamic equations for the system of moments of the leading term of the asymptotic solution are obtained within the framework of the previously developed method of semiclassical asymptotics in the weak diffusion approximation. An illustrative example is considered, for which approximate solutions of the dynamic system of moments are constructed using analytical and numerical methods for various values of the fractional derivative parameter α.

In this work, an experimental and model analysis of the copper bromide vapor active media excitation efficiency was carried out at a fixed energy input and pump power. The experimental results showed that for typical pumping parameters of CuBr + Ne + HBr active elements, the requirements for the excitation source in terms of the voltage amplitude on the storage capacitor can be reduced. It was shown that the reduce of voltage amplitude from 11.3 to 6.6 kV while maintaining the energy input at the level of 1 mJ/cm3, leaded to the generation power decreased by 15

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.

We propose the semi-empirical mathematical model of the brightness amplifier based on the self-terminating transitions in laser active media. The model is applied to the copper bromide vapor brightness amplifier. The feature of the model is that it allows one to distinguish the amplified spontaneous emission and amplified input optical signal. Due to that we have managed to study the “real” gain and theoretical maximum of contrast for the brightness amplifier depending on temporal and energy characteristics of the input optical signal.

We study the leading term of asymptotics for a solution to the two-dimensional kinetic equation with a nonlocal cubic nonlinearity that is a model of the ionization of active medium. The asymptotic solution under consideration is constructed analytically in the class of trajectory concentrated function within the approximation of weak diffusion. The numerical solution to the model equation corresponding to the asymptotic solution is constructed. Using the analytical and numerical methods, the dependence of the residual of the asymptotic solution on time and asymptotic small parameter is studied. The consistency of asymptotic and numerical solutions is ascertained for the studied range of the equation parameters.

We study a non-typical excitation mode of the copper bromide active medium using the kinetic model. The active medium is pumped by the pulse train and the laser generation is obtained in the subsequent single excitation pulse. Such mode allows one to obtain the laser generation pulse with the extended duration by increase in the pause duration after the pulse train. The relaxation processes during the pause are studied to explain such effect. It is shown that this operation mode can also be used to obtain the superradiance and amplification pulses of the extended duration that is of interest for active optical systems. Based on the comparison with the experimental results, new fundamental results are obtained regarding the copper bromide kinetics in the active medium.

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability-generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem.

This paper addresses the results of experimental and model studies of a copper bromide vapor brightness amplifier at high pump pulse repetition rates. The features of the operating mode that are associated with the use of a reduced energy input into the discharge to obtain superradiance and amplification at frequencies above 100 kHz are noted. For the first time, for active media on metal vapors, the superradiance obtained at a pump pulse repetition rate of up to 300 kHz in a CuBr-vapor medium. A prototype model of a high-speed brightness amplifier has been developed.

Nonlocal versions of the reaction-diffusion type population equations can describe the evolution of spatiotemporal structures (patterns) depending on the equation parameter domain. Under conditions of weak diffusion, numerical methods have been used to compare the processes of spatiotemporal pattern formation in a nonlocal population model described by a one-dimensional generalized Fisher–Kolmogorov–Petrovsky–Piskunov equation with nonlocal competitive losses and in a two-dimensional nonlocal version of the kinetic model of quasi-neutral plasma of metal vapor active media described by the kinetic equation with nonlocal cubic nonlinearity. The effect of relaxation on the pattern formation is studied.

We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equation. The family of asymptotic solutions to the kinetic equation is constructed using the symmetry operators acting on functions concentrated in a neighborhood of a point determined by the dynamical system. Based on these solutions, we introduce the nonlinear superposition principle for the nonlinear kinetic equation. Our formalism based on the Maslov germ method is applied to the Cauchy problem for the specific two-dimensional kinetic equation. The evolution of the ion distribution in the kinetically enhanced metal vapor active medium is obtained as the nonlinear superposition using the numerical–analytical calculations.

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

Amplifying characteristics of the copper bromide vapor active media with the increased inversion duration are studied using the detailed kinetic modeling. The analysis of the gain radial profile and its time evolution at various points of the GDT profile is presented. The results show the possibility of application such active media to the tasks of the remote object visualization.

The results of the development of a high-frequency pumping source for active media on self-terminating transitions in metal vapors, which allows operation in the mode of a low energy deposition into the discharge, are presented. Reduced energy deposition into the discharge is provided due to the pumping of the active medium with short-duration high-voltage pulse (3 kV, 15 А, 40–60 ns). A record-high radiation-pulse repetition rate of 200 kHz in the active medium of copper bromide vapors was obtained when operating in the superradiance mode.

This paper presents the theoretical study of the optical gain of copper vapor active media for a wide range of input signal power. The modified kinetic model is used to identify the effect of the optical saturation on the amplified spontaneous emission and the spatio-temporal evolution of amplifying characteristics. Also, the dependence of these characteristics on the copper concentration is studied. The results obtained are discussed in the context of using copper vapor active media in laser monitors.

We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.