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At least 1,063 records · Page 59
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

A Discrete Nonlinear Schrödinger-Type Hierarchy, Its Finite-Dimensional Reduction Analysis, and the Numerical Integration Scheme

We investigate the procedures of discretization of the integrable nonlinear Schrödinger dynamical system, well known as the Ablowitz–Ladik equation, the corresponding symplectic structures, and the finite-dimensional invariant reductions. We develop an efficient scheme of invariant reduction of the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrence algebraic regular relations that allows one to generate solutions of the discrete nonlinear Schrödinger dynamical system and discuss the related functional spaces of solutions. Finally, we analyze the Fourier-transform approach to the study of the set of solutions of the discrete nonlinear Schrödinger dynamical system and its functional-analytic aspects.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

An alternative approach for MLE calculation in nonlinear continuous dynamic systems

As an important metric to tell if a nonlinear dynamic system has a singular attractor or divergent trajectory, the maximal Lyapunov exponent (MLE) can be calculated from either system models or time series of state variable measurement. However, in the real world, due to inaccurate models, measurement noise, and the fact that sometimes state variables cannot be measured directly, it is very difficult to get an accurate MLE, which limits its application in, for example, in prediction of a nonlinear physical system (e. g. power systems, etc.) behavior. To overcome these factors, this paper proposed a trajectory-estimation-based MLE calculation approach. The proposed approach addressed how to calculate the MLE when state variables cannot be accessed directly, and uncertainties in system models, as well as noise in measurements. The simulation results shows that the proposed approach is able to handle well the nonlinear measurement functions between state variables and measurements, and get better results than pure model-based approaches or measurement-based approaches in front of measurement noise and model uncertainties.

maximal Lyapunov exponents, nonlinear differential↗
DOE OSTI2018

Optical soliton perturbation with full nonlinearity by extended trial function method

The extended trial function scheme is applied to retrieve soliton solutions to the perturbed nonlinear Schrödinger’s equation with ten forms of fiber nonlinearity. Bright, dark and singular soliton solutions are retrieved that appear with their respective existence criteria. Some additional forms of nonlinear waves naturally emerge as a byproduct of the integration scheme.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Exact optical solitons in metamaterials with anti-cubic law of nonlinearity by Lie group method

In this paper, using Lie point symmetry analysis, we study the dynamics of the solitons for the nonlinear Schrodinger’s equation with anti-cubic nonlinearity. Some new doubly periodic solutions are obtained that degenerate to dark and bright soliton solutions. In our of best knowledge, the obtained solutions are new. Those obtained results have important applications in the understanding the nonlinear propagation theory of solitons in metamaterials.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Effect of electron–electron interactions on optical properties of GaN/AlN quantum wells: a nonlinear Schrödinger equation approach

Researchers usually neglect the electron–electron interaction effect when they study the optical properties of semiconducting nanostructures through the compact density matrix approach. In the existing papers, this work has also been done through self-consistent solution of the Schrödinger and Poisson equations. For the first time, we have investigated the electronic and optical properties of wurtzite GaN/AlN quantum wells in the presence of electron–electron interactions assuming weak electron–electron scattering through a numerical solution of a nonlinear Schrödinger equation. It is shown that the electron–electron interaction effect leads to the removal of the N-fold degeneracy in the energy levels of N-well multiple quantum well systems. Effects of different parameters such as electric field, the strength of the electron–electron interaction, a total length of the system and also two different numbers of wells (1 and 2) on the mentioned optical properties have been investigated. Also, we observed redshifts in the optical rectification coefficient peak positions when the nonlinear parameter increased. Finite difference self-consistent method was employed to solve the nonlinear Schrödinger equation and we demonstrated the method convergence. Finally, we illustrated the accuracy of the method by comparing our energy values with those of the Sinc method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Competing effects of Kerr nonlinearity and K-photon absorptions on continuous-wave laser inscriptions

Material processing with laser offers a reliable tool in a large variety of micromachining technology, ranging from on-disc inscriptions to cell ablations through DNA combing and imprinting for fabrications of micro and nanofluidic devices. In these applications lasers are designed to operate in specific regimes characterized by their powers and wavelengths, so understanding characteristic properties of the distinct possible laser operation regimes turns out to be a relevant step toward optimization of their uses as well as improvement of the technology. In this work we examine the stability of a model of femtosecond laser intended for laser inscriptions in nonlinear transparent media, taking into consideration the laser-induced material damage and multiphoton ionization. The model approximates the laser dynamics by a complex Ginzburg–Landau equation with a Kerr plus a high-order nonlinear term accounting for K-photon ionization, coupled to a Drude equation for time evolution of the electron plasma density. Following the Benjamin–Feir instability theory it is shown that depending on characteristic parameters of the model, multiphoton ionization processes can stabilize continuous-wave or pulse regimes. These parameters include the group-delay dispersion, the linear and nonlinear gains, the phase shift per roundtrip and the plasma ionization rate.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

A simple design of highly birefringent and nonlinear photonic crystal fiber with ultra-flattened dispersion

We present a compressed hexagonal photonic crystal fiber (CH-PCF) based on a simple structure with ultra-flattened dispersion, high birefringence and nonlinearity. The propagation characteristics are studied accurately by utilizing full vector finite-element method (FEM). Numerical results indicate that the birefringence of 1.59 × 10{sup −2} was obtained with high nonlinear coefficient of 42.58 W{sup −1} km{sup −1}(X) and 52.80 W{sup −1} km{sup −1}(Y) at an effective wavelength of 1.55 μm. Furthermore, CH-PCF presents an ultra-flattened dispersion with variation of ± 3.4 ps/nm/km(X) and ± 1.6 ps/nm/km(Y) over 1.3 μm to 1.8 μm (0.5 μm bandwidth), respectively. In addition, this circular-hole structure can be fabricated conveniently at most easy compared with complex designs, which is an advantage for practical applications such as dispersion compensation and nonlinear optics application.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Nonlinear Polarization Effects in Dielectrics with Hydrogen Bonds

Using quasiclassical kinetic theory, nonlinear phenomena under the formation of space-charge polarization in crystals with hydrogen bonds (HBC) are investigated. From the solution of the nonlinear system of Fokker-Planck and Poisson equations with blocking electrodes, it is established that for the mathematical description of the relaxation polarization in HBCs in weak fields (100-1000 kV/m) and at high temperatures (T> 350 K), it is sufficient to use a linear approximation of perturbation theory. Theoretically, it was found that at the first odd frequency of the alternating field, nonlinear effects caused by the interaction of relaxation modes of various frequency orders start to appear. The diffusion and mobility coefficients are calculated taking into account both mechanisms of the protons transitions (thermally activated and tunneling transitions) through a parabolic potential barrier. A recurrence expression is constructed for calculating complex amplitudes of relaxation modes generated at an arbitrary frequency (multiple to the fundamental frequency) of the alternating field in an arbitrary approximation of perturbation theory. The proposed scheme for solving the kinetic equation can be applied to other crystals with ionic conductivity similar to HBCs in the type and properties of the crystal lattice.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

Detecting dynamic causal inference in nonlinear two-phase fracture flow

Identifying dynamic causal inference involved in flow and transport processes in complex fractured-porous media is generally a challenging task, because nonlinear and chaotic variables may be positively coupled or correlated for some periods of time, but can then become spontaneously decoupled or non-correlated. In his 2002 paper (Faybishenko, 2002), the author performed a nonlinear dynamical and chaotic analysis of time-series data obtained from the fracture flow experiment conducted by Persoff and Pruess (1995), and, based on the visual examination of time series data, hypothesized that the observed pressure oscillations at both inlet and outlet edges of the fracture result from a superposition of both forward and return waves of pressure propagation through the fracture. In the current paper, the author explores an application of a combination of methods for detecting nonlinear chaotic dynamics behavior along with the multivariate Granger Causality (G-causality) time series test. Based on the G-causality test, the author infers that his hypothesis is correct, and presents a causation loop diagram of the spatial-temporal distribution of gas, liquid, and capillary pressures measured at the inlet and outlet of the fracture. The causal modeling approach can be used for the analysis of other hydrological processes, for example, infiltration and pumping tests in heterogeneous subsurface media, and climatic processes, for example, to find correlations between various meteorological parameters, such as temperature, solar radiation, barometric pressure, etc.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

Here, the classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A note on the reliability of goal-oriented error estimates for Galerkin finite element methods with nonlinear functionals

Here, we consider estimating the discretization error in a nonlinear functional J (u) in the setting of an abstract variational problem: find u ϵ $\mathscr{V}$ such that B (u, φ) = L (φ) ∀φ ϵ $\mathscr{V}$, as approximated by a Galerkin finite element method. Here, $\mathscr{V}$ is a Hilbert space, B (. , .) is a bilinear form, and L (∙) is a linear functional. We consider well-known error estimates η of the form J (u) - J (u h ) ≈ η = L (z) - B (u h , z), where u h denotes a finite element approximation to u, and z denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution z. An estimate η is said to be reliable if there exists a constant C ϵ $\mathbb{R}$ >0 independent of u h such that |J (u) - J (u h )| ≤ C|η|. We present several example pairs of bilinear forms and nonlinear functionals where reliability of η is not achieved.

A posteriori↗
DOE OSTI2013

Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves

The nonlinear propagation of planar and nonplanar (cylindrical and spherical) ion-acoustic waves in an unmagnetized electron–positron–ion–dust plasma with two-electron temperature distributions is investigated in the context of the nonextensive statistics. Using the reductive perturbation method, a modified nonlinear Schrödinger equation is derived for the potential wave amplitude. The effects of plasma parameters on the modulational instability of ion-acoustic waves are discussed in detail for planar as well as for cylindrical and spherical geometries. In addition, for the planar case, we analyze how the plasma parameters influence the nonlinear structures of the first- and second-order ion-acoustic rogue waves within the modulational instability region. The present results may be helpful in providing a good fit between the theoretical analysis and real applications in future spatial observations and laboratory plasma experiments. -- Highlights: ► Modulational instability of ion-acoustic waves in a new plasma model is discussed. ► Tsallis’s statistics is considered in the model. ► The second-order ion-acoustic rogue wave is studied for the first time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Investigation of odd-order nonlinear susceptibilities in atomic vapors

We theoretically deduce the macroscopic symmetry constraints for arbitrary odd-order nonlinear susceptibilities in homogeneous media including atomic vapors for the first time. After theoretically calculating the expressions using a semiclassical method, we demonstrate that the expressions for third- and fifth-order nonlinear susceptibilities for undressed and dressed four- and six-wave mixing (FWM and SWM) in atomic vapors satisfy the macroscopic symmetry constraints. We experimentally demonstrate consistence between the macroscopic symmetry constraints and the semiclassical expressions for atomic vapors by observing polarization control of FWM and SWM processes. The experimental results are in reasonable agreement with our theoretical calculations. -- Highlights: •The macroscopic symmetry constraints are deduced for homogeneous media including atomic vapors. •We demonstrate that odd-order nonlinear susceptibilities satisfy the constraints. •We experimentally demonstrate the deduction in part.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Enhanced diode-like operation mediated by an asymmetric non-Hermitian and nonlinear defect

In this work we study the nonreciprocal transport along two tight-binding chains which are connected by a single defect having an asymmetric non-Hermitian nonlinear off-diagonal coupling. The spectrum of transmission and reflection, the gain curve and the rectifying factor are analytically obtained using a backward iterative process. A set of discrete linear Schroedinger equations is used to model the wave propagation through the two Hermitian side chains, while a discrete nonlinear Ablowitz–Ladik equation governs their coupling by the single defect. We show that the emergence of a multistability window induced by the non-linear contribution, together with the biased transport promoted by a parity-breaking non-Hermiticity, generates an efficient rectification of the transmitted wave component. - Highlights: • Transmission along a quantum channel with a defect is analytically studied. • The defect has non-Hermitian, nonlinear and non-symmetric characteristics. • The transmission spectrum exhibits multistability regions. • The non-conventional properties of the defect lead to a diode-like behavior. • Multistability is shown to promote an enhanced rectification of the transmitted signal.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: Generalized coherent states for nonlinear algebras

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

We formulate and study dynamics from a complex Ginzburg–Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg–Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg–Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrödinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). We also find a collection of transient features, including transient defects and what appear to be rogue waves, while in two spatial dimensions we observe highly localized pattern formation. While saturation will often regularize the dynamics, such transient dynamics can still be observed – and in some cases even prolonged – as the saturability of the media is increased, as the saturation may act to slow the timescale.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Nonlinear dynamic analysis of solution multiplicity of buoyancy ventilation in a typical underground structure

Buoyancy ventilation is widely used in underground buildings, such as underground hydropower stations. Multiple solutions of buoyancy ventilation may exist in those underground structures. In this study, we developed a transient model comprising an ordinary differential equation system to describe buoyancy ventilation patterns in typical two-zone underground structures. Additionally, the accuracy of the model was validated. Nonlinear dynamical analysis was conducted to study multiple steady-state airflow. According to mathematical derivation, the configuration of one local heat source at the bottom corner introduces two stable solutions. The criterion to determine the stability and existence of solutions for more general scenarios was developed. Using this criterion, we obtained the multiple steady states of any two-zone underground buildings for different stack height ratios and the strength ratios of the heat sources. Furthermore, this criterion can be adopted for the design of buoyancy ventilation or natural smoke ventilation systems. Designers can change the height ratio of the stack or the heat ratio of two zones to induce the desired ventilation patterns. Finally, a case study was conducted with field measurements to demonstrate the use of the nonlinear dynamical analysis method to investigate the multiple steady states of buoyancy ventilation. Through the case study, we validated that the proposed criterion could produce the same result as the nonlinear dynamical analysis.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2018

Biomimetic model-based advanced control strategy integrated with multi-agent optimization for nonlinear chemical processes

Here, a novel framework is proposed for integrating biomimetic-based advanced control and multi-agent optimization approaches for nonlinear chemical process applications. In particular, a Biologically-Inspired Optimal Control Strategy, denoted as BIO-CS, is combined with multi-agent optimization (MAO) techniques to provide optimal solutions for dynamic systems. In this combined framework, the BIO-CS algorithm employs gradient-based optimal control solvers for the intermediate problems associated with the leader-follower agents’ local interactions. Also, the MAO uses the capabilities of heuristic-based optimization techniques by sharing process information to obtain optimal operating setpoints for the controller considering an overall process objective. The applicability of the proposed method is demonstrated using a nonlinear, multivariable, process model of a fermentation system. Specifically, the optimal operating points are computed by the MAO implementation for setpoint tracking, trajectory tracking and plant-model mismatch scenarios for BIO-CS application. Results of the developed framework are compared to a gradient-based Sequential Quadratic Programming (SQP) technique and a classical proportional-integral (PI) controller in terms of optimization and control studies, respectively. As an additional contribution, BIO-CS is also cast as a model predictive controller (MPC) for the first time and compared to the agent-based BIO-CS approach in terms of computational time and tracking error. Closed-loop control results show up to 46% improvement in tracking performance during transient for the multi-agent BIO-CS when compared to BIO-CS as MPC for additional computational expense. The obtained results illustrate the capabilities of this novel integrated framework including BIO-CS as MPC to achieve desired nonlinear system performance for various scenarios.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗