Taras I. Lakoba

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrodinger equations with periodic potentials, and isolated solitons in GinzburgLandau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.

All-optical multichannel 2R regeneration in a fiber-based device

We propose the design of an all-optical 2R regenerator capable of handling multiple wavelength-division-multiplexed channels simultaneously. It extends the known concept of off-center filtering of self-phase-modulation-broadened signal spectra. The novel feature of the proposed device is a dispersion map that strongly suppresses interchannel impairments. The map employs several sections of nonlinear fiber with high normal dispersion, separated by dispersion compensators with spectrally periodic group delay. The results of our numerical simulations indicate the feasibility of such a multichannel regenerator.

A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

The Petviashvilis iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: -Mu+up=0, where M is a positive definite self-adjoint operator and p=const. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.

A new robust regime for a dispersion-managed multichannel 2R regenerator

We study the performance of a multichannel version [M. Vasilyev and T.I. Lakoba, Opt. Lett. 30, 1458 (2005)] of the all-optical Mamyshev regenerator in a practically important situation where one of its key components - a periodic-group-delay device - has a realistic amplitude characteristic of a bandpass filter. We show that in this case, the regenerator can no longer operate in the regime reported in our original paper. Instead, we have found a new regime in which the regenerators performance is robust not only to such filtering, but also to considerable variations of regenerator parameters. In this regime, the average dispersion of the regenerator must be (relatively) large and anomalous, in constrast to what was considered in all earlier studies of such (single-channel) regenerators based on spectral broadening followed by off-center filtering. In addition, hardware implementation of a regenerator in the new regime is somewhat simpler than that in the original regime.

Identifying Useful Statistical Indicators of Proximity to Instability in Stochastic Power Systems

Prior research has shown that autocorrelation and variance in voltage measurements tend to increase as power systems approach instability. This paper seeks to identify the conditions under which these statistical indicators provide reliable early warning of instability in power systems. First, the paper derives and validates a semi-analytical method for quickly calculating the expected variance and autocorrelation of all voltages and currents in an arbitrary power system model. Building on this approach, the paper describes the conditions under which filtering can be used to detect these signs in the presence of measurement noise. Finally, several experiments show which types of measurements are good indicators of proximity to instability for particular types of state changes. For example, increased variance in voltages can reliably indicate both proximity to a bifurcation and the location of increased stress. On the other hand, growth of autocorrelation in certain line currents is related less to a specific location of stress but, rather, is a reliable indicator of stress occurring somewhere in the system; in particular, it would be a clear indicator of approaching instability when many nodes in an area are under stress.

Understanding Early Indicators of Critical Transitions in Power Systems From Autocorrelation Functions

Many dynamical systems, including power systems, recover from perturbations more slowly as they approach critical transitions - a phenomenon known as critical slowing down. If the system is stochastically forced, autocorrelation and variance in time-series data from the system often increase before the transition, potentially providing an early warning of coming danger. In some cases, these statistical patterns are sufficiently strong, and occur sufficiently far from the transition, that they can be used to predict the distance between the current operating state and the critical point. In other cases CSD comes too late to be a good indicator. In order to better understand the extent to which CSD can be used as an indicator of proximity to bifurcation in power systems, this paper derives autocorrelation functions for three small power system models, using the stochastic differential algebraic equations (SDAE) associated with each. The analytical results, along with numerical results from a larger system, show that, although CSD does occur in power systems, its signs sometimes appear only when the system is very close to transition. On the other hand, the variance in voltage magnitudes consistently shows up as a good early warning of voltage collapse.

Multi-Wavelength All-Optical Regeneration

We discuss the theoretical and experimental progress in the development of multi- wavelength all-optical 2R regenerators, with emphasis on innovative dispersion management in the scheme based on off-center filtering of self-phase-modulation-broadened signal spectrum (Mamyshev regenerator).

Multicanonical Monte Carlo Study of the BER of an All-Optically 2R Regenerated Signal

Using a numerical procedure based on the multicanonical Monte Carlo (MMC) algorithm, we compute the bit error rate (BER) at the output of a Mamyshev-type all-optical regenerator. For the specific device considered, this BER is degraded as compared to its value at the input to the regenerator, even though the -factor is improved. In absolute terms, the output BER decreases when the bandwidth of the optical filter placed before the regenerator to limit the amount of entering noise is decreased. We also present evidence that the degradation of the BER caused by the regenerator is due to high sensitivity of the output signal to small variations of the input signal shape. To our knowledge, this fact has not been pointed out in earlier studies of regenerators. In addition to the previous results, the modification of the MMC procedure that we proposed and used in this paper can be, in a certain sense, parallelized. When it is practical to do so, our procedure provides significant saving of the computational time over the standard MMC simulation.

Multichannel all-optical regeneration

We present our experimental results on 12×10 Gb/s all-optical 2R regeneration, enabled by the group-delay management scheme in Mamyshev regenerator, and discuss extension of this approach beyond 2R regeneration.

Conjugate Gradient method for finding fundamental solitary waves

Abstract The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it can find fundamental solitary waves of nonlinear Hamiltonian equations. The main obstacle that such a modified CGM overcomes is that the operator of the equation linearized about a solitary wave is not sign definite. Instead, it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum. We present versions of the modified CGM that can find solitary waves with prescribed values of either the propagation constant or power. We also extend these methods to handle multi-component nonlinear wave equations. Convergence conditions of the proposed methods are given, and their practical implications are discussed. We demonstrate that our modified CGMs converge much faster than, say, Petviashvili’s or similar methods, especially when the latter converge slowly.