In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.

Предложен метод численного решения четырехволновых кинетических уравнений, возникающих в задачах волновой (слабой) турбулентности при описании однородного изотропного взаимодействия волн. Для расчета интеграла столкновений разработаны быстросходящиеся кубатурные формулы, позволяющие адаптировать алгоритм к особенностям решений и ядер интегралов. Проведены эксперименты на сходимость в задачах интегрирования из реальных приложений. Для учета существенной разномасштабности задач турбулентности в алгоритме реализованы и протестированы дробно-рациональные приближения решений и новая схема итераций по времени. Эффективность разработанного алгоритма продемонстрирована при моделировании обратного каскада частиц бозе-газа при формировании конденсата Бозе–Эйнштейна. Библ. 51. Фиг. 10. Табл. 1.

The general characteristics of an optical signal as a result of generation in a resonator can be described using a dynamic model based on the complex cubic Ginzburg–Landau equation, which takes into account the saturated gain and dissipative terms responsible for the distributed action of various intracavity devices. The paper proposes two new effective modifications of the split-step Fourier method for a numerical solution to equations of this type. The first algorithm is based on the application of a new way of separating physical processes affecting the optical signal during propagation in a fiber, which made it possible to express the action of both nonlinear and dispersive spatial steps by explicit analytical expressions. The second proposed method enabled significant improvement in the accuracy of calculations due to including energy evolution in the coefficients of the equation. Numerical experiments have shown that the new schemes can produce the second order of approximation with respect to the evolutionary variable in contrast to the classical scheme that provides only the first order of approximation.

We propose a high precision algorithm for solving the Gelfand–Levitan–Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson’s type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations. To the best of our knowledge, this is the first algorithm to solve this problem with an order of accuracy higher than the second.

We develop a numerical method for solving kinetic equations (KEs) that describe out-of-equilibrium isotropic nonlinear four-wave interactions in optics, deep-water wave theory, physics of superfluids and Bose gases, and in other applications. High complexity of studying numerically the wave kinetics in these applications is related with the multi-scale nature of turbulence and with power-law behaviour of turbulent spectra in the Fourier space. When solving the Cauchy problem for KE, this leads to emergence of spectra with extremely steep gradients and to occurrence of singular points in the collision integral standing in the right-hand side. To solve these problems, we develop special fast-convergent cubature formulas, highly-accurate rational approximations of the KE solution, and a new stable method for time marching. We apply the developed methods for solving the test problems of integration arisen from applications, for studying the wave kinetics in random fiber lasers and for analysing the Bose–Einstein condensation. In these applications we used KEs obtained from the Ginzburg–Landau and from the Gross–Pitaevskii equations.

The nonlinear Fourier transform (NFT) is an approach that is similar to a conventional Fourier transform. In particular, NFT allows to analyze the structure of a signal governed by the nonlinear Schrödinger equation (NLSE). Recently, NFT applied to NLSE has attracted special attention in applications of fiber-optic communication. Improving the speed and accuracy of the NFT algorithms remains an urgent problem in optics. We present an approach that allows to find all variants of symmetric exponential splitting schemes suitable for the fast NFT (FNFT) algorithms with low complexity. One of the obtained schemes showed good numerical results in computing the continuous spectrum compared with other fast fourth-order NFT schemes.

The nonlinear Fourier transform (NFT) is a method for analyzing signal structures influenced by the nonlinear Schrödinger equation (NLSE). Enhancing the performance and accuracy of NFT algorithms is crucial in fiber optics. We propose a method to identify all suitable symmetric exponential splitting schemes that enhance the speed of fast NFT (FNFT) algorithms while maintaining low numerical complexity. The schemes we developed performed well in numerical tests, comparing favorably with other fast 4th order NFT algorithms.

We propose the method for numerical solution of four-wave kinetic equations that arise in the wave turbulence (weak turbulence) theory when describing a homogeneous isotropic interaction of waves. To calculate the collision integral in the right-hand side of equation, the cubature formulas of high rate of convergence are developed, which allow for adaptation of the algorithm to the singularities of the solutions and of the integral kernels. The convergence tests in the problems of integration arising from real applications are done. To take into account the multi-scale nature of turbulence problems in our algorithm, rational approximations of the solutions and a new time marching scheme are implemented and tested. The efficiency of the developed algorithm is demonstrated by modelling the inverse cascade of Bose gas particles during the formation of a Bose–Einstein condensate.

We propose a high-precision algorithm for solving the three-component Gelfand–Levitan–Marchenko equations (GLME) associated with the Manakov system, which describes the behavior of light waves through the optical fibers. The algorithm generalizes the high-order generalized Toeplitz inner-bordering method for solving the two-component GLME associated with the nonlinear Schrödinger equation. Numerical experiments have shown that the proposed algorithm makes it possible to increase the accuracy of solving the GLME associated with Manakov system up to the sixth order.

We use the concept of gauge transformations in the proof of the invariance of the statistics of zero-vorticity lines in the case of the inverse energy cascade in wave optical turbulence; we study it in the framework of the hydrodynamic approximation of the two-dimensional nonlinear Schrödinger equation for the weight velocity field 𝐮 . The multipoint probability distribution density functions f_n of the vortex field Ω=∇×𝐮 satisfy an infinite chain of Lundgren–Monin–Novikov equations (statistical form of the Euler equations). The equations are considered in the case of the external action in the form of white Gaussian noise and large-scale friction, which makes the probability distribution density function statistically stationary. The main result is that the transformations are local and conformally transform the n -point statistics of zero-vorticity lines or the probability that a random curve 𝐱(l) passes through points 𝐱_i∈ℝ^2 for l=l_i , i=1,…,n , where Ω=0 , is invariant under conformal transformations.

Abstract We propose a generalized method for solving the Gelfand–Levitan–Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.

Optical turbulence is described in terms of multipoint probability density distribution functions (PDF) f n using the Lundgren–Monin–Novikov (LMN) equation (statistical form of the Euler equation) for the field of vortex w = ∇ × u in a 2D flow ( u is the weight velocity field). The evolution of Lagrangian particles occurs along the characteristics of the f n equation from the LMN hierarchy. The vorticity is preserved along the characteristics in the absence of an external random force. It is shown that the G group of conformal transformations invariantly transforms the characteristics of the equation with zero vorticity and the family of f n equations for PDF along these lines, or the statistics of zero-vorticity lines. Along other level lines w = const ≠ 0, the statistics is not conformally invariant. In addition, the action of G conserves the PDF class.

Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for wave turbulence arising in the Nonlinear Schrodinger model and for the gravitational waves in the Einstein's vacuum field model. We show that inverse transfer is not described by a scaling of the constant-flux solution but has an anomalous scaling. We compute the anomalous exponents and analyze their origin using the theory of dynamical systems.

The inverse scattering transform (IST) allows to integrate the nonlinear Schrödinger equation (NLSE) equation analytically [1] and consists of three main steps: the first step is solving the direct Zakharov-Shabat problem (ZSP) to determine scattering data, the second is an evolution of the scattering data, and the third step is solving the inverse scattering problem to restore a solution from the scattering data. This method, also known as the nonlinear Fourier transform (NFT), has recently attracted much attention in areas where NLSE is used to describe various types of optical signals such as lasers and telecommunications.

We propose a new method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB) with an arbitrary point to start the calculation. This makes it possible to find solutions of the GLME at an arbitrary point with a cutoff of the matrix coefficient, which allows to avoid the occurrence of numerical instability and to perform calculations for soliton solutions spaced apart in the time domain. Using an example of two solitons, we demonstrate our method and its range of applicability. An example of eight solitons shows how the method can be applied to a more complex signal configuration.

Production process models are the intellectual core of decision support systems in agriculture. The tasks faced by the users of these systems are quite varied and may require the operation of the same model of the production process in different environments. At the same time, for the practical application of models in solving applied problems, a convenient user interface is required in any environment. The.NET Framework standard library includes the System. Component Model namespace, which describes the abstract user interface entities that are not tied to a specific technology to ensure that the user interface works efficiently in different environments. The RW.Ring platform extends this library with services that provide an interface for working with background tasks, notifying the user with information messages, and managing an application instance. It should be noted that the scope of this development is limited to the client sphere: it is obvious that the architecture of the Web server, which generates html in response to client requests, does not allow creating dialogs from server logic. However, even in distributed applications, as a rule, there is a specialized client where this architecture is quite applicable.

The nonlinear Schrödinger equation is widely used in telecommunication applications, because it allows one to describe the propagation of pulses in an optical fiber. Recently some new approaches based on the nonlinear Fourier transform (NFT) have been actively explored to compensate for fiber nonlinearity and to exceed the limitations of nonlinearity-imposed limits of linear transmission methods. The first step in the NFT method is the solution of the direct scattering problem for the Zakharov-Shabat (ZS) system. Improving the accuracy of computational methods to solve the direct ZS problem remains an urgent problem in optics. In particular, it is important to increase the approximation order of the methods, especially in problems where it is necessary to analyze the structure of complex waveforms. In addition multi-soliton pulses are potential candidates for fiber optical transmission, where the information is modulated and recovered in the so-called nonlinear Fourier domain. To correctly describe them and their spectral parameters, more accurate and fast numerical methods are needed.