Nonlinear Theory Research Articles

Near-term distributed quantum computation using mean-field corrections and auxiliary qubits

Distributed quantum computation is often proposed to increase the scalability of quantum hardware, as it reduces cooperative noise and requisite connectivity by sharing quantum information between distant quantum devices. However, such exchange of quantum information itself poses unique engineering challenges, requiring high gate fidelity and costly non-local operations. To mitigate this, we propose near-term distributed quantum computing, focusing on approximate approaches that involve limited information transfer and conservative entanglement production. We first devise an approximate distributed computing scheme for the time evolution of quantum systems split across any combination of classical and quantum devices. Our procedure harnesses mean-field corrections and auxiliary qubits to link two or more devices classically, optimally encoding the auxiliary qubits to both minimize short-time evolution error and extend the approximate scheme’s performance to longer evolution times. We then expand the scheme to include limited quantum information transfer through selective qubit shuffling or teleportation, broadening our method’s applicability and boosting its performance. Finally, we build upon these concepts to produce an approximate circuit-cutting technique for the fragmented pre-training of variational quantum algorithms. To characterize our technique, we introduce a non-linear perturbation theory that discerns the critical role of our mean-field corrections in optimization and may be suitable for analyzing other non-linear quantum techniques. This fragmented pre-training is remarkably successful, reducing algorithmic error by orders of magnitude while requiring fewer iterations.

Controlling lasing around exceptional points in coupled nanolasers

Coupled nanolasers are of growing interest for on-chip optical computation and data transmission, which requires an understanding of how lasers interact to form complex systems. The non-Hermitian interaction between two coupled resonators, when excited selectively, can lead to parity-time symmetry, the formation of exceptional points, and subsequently spectral control and increased sensitivity. These investigations have been limited to pump energies close to the lasing threshold, and large or narrow-line lasers. Here, by programmable optical excitation we study two coupled nanolasers significantly above threshold, where mode instability plays an important role. We map the mode evolution around two exceptional points, and observe lasing gaps due to reversed pump dependence which compare well with non-linear theory. Finally, the coupling can be exploited to control the lasing threshold and wavelength, and for frequency switching around the lasing gap. Controlled and integrated nanolasers constitutes a promising platform for future highly sensitive and programmable on-chip laser sources.

A Non-linear Analysis of Segmental Harmony in Algerian Wadi Souf Dialect

The present study discusses some phonological processes of segmental harmony in Algerian Wadi Souf dialect. The data were collected from natural recorded speech of fifty native speakers of this dialect. Moreover, one of the researchers added some examples since she is a native speaker of this dialect. This research adopts the non-linear phonological theory. The findings showed that this theory can provide an adequate analysis for segmental harmony. It is also found that non-adjacent sibilants can spread the feature of anteriority bi-directionally. The anterior sibilants /z/ and /s/ spread the feature [+anterior] to the non-anterior sibilant /ʒ/, and it is realized as [z].  Moreover, raising and fronting vowel harmony occurs in both CaCi:C adjectives that are realized as CiCi:C and the imperfect prefix of jaCCiC and jaCCi verbs that are realized as jiCCiC and jiCCi. Raising and rounding vowel harmony occurs in the imperfect prefix of jaCCuC tri-consonantal verbs that are realized as juCCuC.

Connecting weakly nonlinear elasticity theories of isotropic hyperelastic materials

Soft materials exhibit significant nonlinear geometric deformations and stress–strain relationships under external forces. This paper explores weakly nonlinear elasticity theories, including Landau’s and Murnaghan’s formulations, advancing understanding beyond linear elasticity. We establish connections between these methods and extend strain-energy functions to the third and fourth orders in power of [Formula: see text], where [Formula: see text] and [Formula: see text], and [Formula: see text] is the perturbation to the deformation gradient tensor [Formula: see text]. Furthermore, we address simplified strain-energy functions applicable to incompressible materials. Through this work, we contribute to a comprehensive understanding of nonlinear elasticity and its relationship to weakly nonlinear elasticity, facilitating the study of moderate deformations in soft material behavior and its practical applications.

Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

In a tokamak fusion reactor operated at steady state, the equilibrium magnetic field is likely to have reversed shear in the core region, as the noninductive bootstrap current profile generally peaks off-axis. The reversed shear Alfvén eigenmode (RSAE) as a unique branch of the shear Alfvén wave in this equilibrium, can exist with a broad spectrum in wavenumber and frequency, and be resonantly driven unstable by energetic particles (EP). After briefly discussing the RSAE linear properties in burning plasma condition, we review several key topics of the nonlinear dynamics for the RSAE through both wave-EP resonance and wave-wave coupling channels, and illustrate their potentially important role in reactor-scale fusion plasmas. By means of simplified hybrid MHD-kinetic simulations, the RSAEs are shown to have typically broad phase space resonance structure with both circulating and trapped EP, as results of weak/vanishing magnetic shear and relatively low frequency. Through the route of wave-EP nonlinearity, the dominant saturation mechanism is mainly due to the transported resonant EP radially decoupling with the localized RSAE mode structure, and the resultant EP transport generally has a convective feature. The saturated RSAEs also undergo various nonlinear couplings with other collective oscillations. Two typical routes as parametric decay and modulational instability are studied using nonlinear gyrokinetic theory, and applied to the scenario of spontaneous excitation by a finite amplitude pump RSAE. Multiple RSAEs could naturally couple and induce the spectral energy cascade into a low frequency Alfvénic mode, which may effectively transfer the EP energy to fuel ions via collisionless Landau damping. Moreover, zero frequency zonal field structure could be spontaneously excited by modulation of the pump RSAE envelope, and may also lead to saturation of the pump RSAE by both scattering into stable domain and local distortion of the continuum structure.

Reduced model and nonlinear analysis of localized instabilities of residually stressed cylinders under axial stretch

In this paper, we present a dimensional reduction to obtain a one-dimensional model to analyze localized necking or bulging in a residually stressed circular cylindrical solid. The nonlinear theory of elasticity is first specialized to obtain the equations governing the homogeneous deformation. Then, to analyze the nonhomogeneous part, we include higher-order correction terms of the axisymmetric displacement components leading to a three-dimensional form of the total potential energy functional. Details of the reduction to the one-dimensional form are given. We focus on a residually stressed Gent material and use numerical methods to solve the governing equations. Two loading conditions are considered. First, the residual stress is maintained constant, while the axial stretch is used as the loading parameter. Second, we keep the pre-stretch constant and monotonically increase the residual stress until bifurcation occurs. We specify initial conditions, find the critical values for localized bifurcation, and compute the change in radius during localized necking or bulging growth. Finally, we optimize material properties and use the one-dimensional model to simulate necking or bulging until the Maxwell values of stretch are reached.

Free vibration analysis of the geometrically nonlinear functionally graded porous curved panels in deterministic and stochastic domains considering various boundary conditions

This study focuses on the analysis of free vibrations in functionally graded (FG) panels with characteristics such as porosity, curvature, and geometric nonlinearity. The analysis encompasses both deterministic and stochastic domains while considering a range of boundary conditions. The nonlinear finite element formulation adopted in this study is based on a higher-order nonlinear shell theory that incorporates the von Karman geometric nonlinearity. The material properties of the FG curved panels are graded using a power law distribution. The investigation includes four different boundary conditions and various types of curved panels, namely spherical, hyperbolic paraboloidal, elliptical, and cylindrical. Stochastic analysis employs the first-order perturbation technique (FOPT), while Monte Carlo simulation (MCS) is employed solely for validating the FOPT model. The study explores the influence of boundary conditions, panel types, porosity distribution, volume fraction index, and other factors in both deterministic and stochastic domains, taking into account randomness in material properties. The stochasticity in the geometric properties like dimensions, and boundary conditions can be the potential future scope of this study.

Spatial energy evolution of focused waves generated in numerical wave tank

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.

Analytical solution to second-harmonic generation of ultrashort laser pulse

We present an analytical method to evaluate the second-harmonic generation (SHG) process of ultrashort laser pulses in a nonlinear medium within the small-signal regime. Building upon the broadband nonlinear coupled wave theory, we derive an analytical solution that allows one to investigate the fundamental physics and detailed dynamics of the nonlinear interaction between ultrashort laser pulses. We find that SHG for ultrashort laser pulses in a nonlinear crystal arises from a series of self-SHG and self sum-frequency generation processes involving different frequency components. We have applied the analytical method to investigate specific pulse profiles with Gaussian and sech functions and found good agreement between analytical theory and numerical simulations, validating the accuracy of the analytical approach. The analytical method enables one to explore the evolution of the second-harmonic wave under various pulse profiles, pulse widths, and crystal thicknesses, and offers a more efficient and insightful approach to study the nonlinear optical dynamics of ultrashort laser pulses.

Targeted Energy Transfer Evaluation for Nonlinear Vibrations of Elastic Medium-Finite Length Beam System Using Nonlinear Energy Sink Theory

The elastic medium can usually reduce the vibration of supporting structure, and the impact of the interaction on the vibration characteristics of the structure is similar to the characteristic of nonlinear energy sink. At present, the dynamic research of beam on elastic foundation considering soil motion has been paid more and more attention. According to the modified Winkler model, the finite-depth elastic medium can be considered to the nonlinear energy sink mass, and the vibration energy dissipation capacity and parameter optimization of the elastic medium supporting finite-length beam under half sine pulse are studied. The Galerkin truncation is applied to the discretization of the governing equations. The numerical solution of the beam coupling system with simple support on the elastic medium is obtained by applying the fourth-order Runge–Kutta method. Based on this, the input energy ratio of the elastic medium dissipation is investigated. Furthermore, through the analysis and optimization of targeted energy transfer and dissipation, the dissipation effect of the finite range elastic medium on the vibration energy of its supporting beam is revealed, and the optimal parameter range of the elastic medium is proved. The results show that after adjusting the elastic medium parameters by technical means, the nonlinear energy sink can absorb most of the vibration energy of the beam quickly and effectively, and the optimal energy dissipation ratio can reach 95.16[Formula: see text]. The quantitative evaluation of the energy dissipation in elastic medium within soil–structure interaction effect is realized.

Nonlinear Stability and Asymptotic Behavior of Periodic Wave Trains in Reaction–Diffusion Systems Against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

Research on two stage obstacle-avoidance trajectory planning and trajectory tracking control in curves

The existing obstacle-avoidance trajectory planning and trajectory tracking control algorithms have limitations such as long-time consumption, high failure rate in dynamic traffic environments, and insufficient trajectory tracking accuracy in curved roads. Based on the above problems, this paper designs a two stage obstacle-avoidance trajectory planner based on nonlinear optimization theory. In first stage Part-NLP, only considering the safety obstacle avoidance, a point mass model and linearization constraints are established to quickly solve the initial trajectory. In the second stage Full-NLP, considering smooth soft constraints comprehensively, the initial trajectory is optimized by establishing driving corridors and a lightweight iterative framework. In control module, this paper selects a linear quadratic form lateral trajectory tracking controller, and the parameters were optimized through the carnivorous plant algorithm. The joint simulation results show that in dynamic traffic environment of curved roads, the two stage planner proposed can accurately plan safe and smooth obstacle avoidance trajectories, and there is a significant reduction in time consumption compared to traditional NLP algorithms. The control strategy can accurately track the planned trajectories, with lateral error controlled within plus or minus 0.1 m, heading error controlled within plus or minus 0.15 rad, speed tracking error controlled within plus or minus 0.15 m/s, and vehicle yaw angle error controlled within plus or minus 0.04 rad; the hardware-in-loop test results indicate that the controller can achieve real-time and accurate trajectory tracking.

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

We present in this paper four versions of chaotic and hyperchaotic modified nonlinear Schrödinger equations (MNSEs). These versions are hyperchaotic integer order, hyperchaotic commensurate fractional order, chaotic non-commensurate fractional order, and chaotic distributed order MNSEs. These models are regarded as extensions of previous models found in literature. We also studied their dynamics which include symmetry, stability, chaotic and hyperchaotic solutions. The sufficient condition is stated as a theorem to study the existence and uniqueness of the solutions of hyperchaotic integer order MNSE. We state and prove another theorem to test the dependence of the solution of hyperchaotic integer order MNSE on initial conditions. By similar way, we can introduce the previous two theorems for the other versions of MNSEs. The Runge-Kutta of the order 4, the Predictor-Corrector and the modified spectral numerical methods are used to evaluate the numerical solutions for integer, fractional and distributed orders MNSEs, respectively. We calculate numerically using the Lyapunov exponents the intervals of parameters of the purposed models at which hyperchaotic, chaotic and stable solutions are exist. The MNSEs have an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, superconductivity, chemical and biological dynamics, lasers and plasmas. The compound synchronization for these chaotic and hyperchaotic models is investigated. We state its scheme using the tracking control technique among three integer commensurate and non-commensurate orders as the derive models and one distributed order as a slave model. We presented and proved a theorem that provides us with the analytical formula for the control functions which are required to achieve compound synchronization. The analytical results are supported by numerical calculations and agreement is found.

A Constitutive Model for Stud Connection in Composite Structures

The complexity of finite element analysis for composite structures can be significantly reduced by representing the connector and adjacent concrete as a macroscopic element. Nevertheless, the prevailing macroscopic models for shear connections predominantly employ nonlinear elastic theory. This approach introduces inaccuracies in estimating structural stiffness and load-bearing capabilities, primarily due to its inability to precisely capture the cumulative effects of plastic damage. In response, this study introduces a novel macroscopic elastoplastic model grounded in plasticity theory, aimed at accurately characterizing the nonlinear behavior of stud connections subjected to concurrent shear and tensile forces. This paper meticulously delineates the implementation of the elastoplastic constitutive model using the backward Euler method for numerical integration. It further articulates the derivation of the consistent tangent stiffness, which aligns with the convergence efficiency of the Newton–Raphson iterative approach. The computation of the element stiffness matrix for a two-node element is executed via the governing equation inherent to the finite element method. An exemplar macroelement test conducted in ABAQUS affirms the implicit backward Euler scheme’s stability and consistency across varying tolerances. Validation of the elastoplastic model against empirical test outcomes corroborates its efficacy, demonstrating the model’s precision in predicting the load–displacement behavior of stud connections under the influence of shear and tensile forces.

CONSIDERATION OF GEOMETRIC NONLINEARITY IN MATHEMATICAL MODELING OF PROBLEMS OF THE THEORY OF SPRING

Solutions to many problems that are important for practice that arise in modern technology cannot always be obtained by traditional methods of the theory of analytical functions or by means of integral transformations. This applies, for example, to contact problems in which the finite dimensions of the region are taken into account in at least one direction, or media with curvilinear anisotropy are studied, etc. The means of the mathematical theory of elasticity are not very effective for the study of such problems. In this case, it is advisable to use the achievements of the potential theory. The use of asymptotic methods at the same time, even in complex cases, makes it possible to obtain well-founded approximate equations, clarify qualitative regularities, and obtain analytical solutions to problems. This paper presents a generalization of the perturbation method, which makes it possible to reduce the study of complex problems of geometrically nonlinear elasticity theory (in the plane and spatial formulation) to the consistent solution of simpler boundary value problems of the potential theory. The geometrically nonlinear theory of elasticity contains some features that make it different from the classical (linear) theory. The main difference is to take into account the difference between the geometry of the undeformed and deformed states of the body under study, when there are movements that cause significant changes in the geometry of the body. At the same time, the equilibrium equation must be drawn up taking into account changes in the shape and size of structures. Taking into account finite deformations, which when creating mathematical models leads to significant difficulties in solving problems, but at the same time brings the model closer to the real problem.

Contraction Analysis of Multi-Agent Control for Guaranteed Capture of a Faster Evader

Abstract This work presents a verifiable condition for the selection of a sufficient number of pursuers to capture a faster evader. The condition is based on the tracking performance of a multi-agent control scheme. Trajectory tracking results are provided for both the effects of the multi-agent control topology and its execution by the pursuers in the context of input saturation. To that end, nonlinear contraction theory is leveraged because it provides a unifying framework for the analysis of systems subject to bounded disturbances. Monte Carlo simulations are performed to validate the proposed condition for sufficient pursuers selection.

On viscous stratified Darcy–Forchheimer flow in a horizontal porous layer with thermal anisotropy and variable permeability

This study analyzes the effect of anisotropy and the internal heat source in a Darcy–Forchheimer porous layer. It is well known that the variations in viscosity can be attributed to the temperature. Therefore, in the present problem, we consider a linear variation in viscosity with temperature for simplicity. We first derived the linear instability theory and then established global stability using the energy functional approach. In the global stability analysis, we show that working with the L2 norm fails to give a sufficient condition for global stability by exhibiting that the associated maximization problem is unbounded in the underlying stability measure space. Then, we show that a conditional stability bound can be achieved by restricting the internal heat source parameter Q with higher-order norms. The eigenvalue problems obtained in linear and nonlinear theories were integrated numerically. The linear and nonlinear instability thresholds are then compared to identify the potential regions of sub-critical instabilities. It is observed that the system is stabilized when the horizontal component of thermal diffusivity dominates and is unstable when the vertical component of thermal diffusivity dominates. We also found that increasing the variable permeability parameter λ destabilized the system. It is observed that increasing viscosity stabilizes the system, and decreasing viscosity encourages the start of convection. It is also interesting that, in the presence of an internal heat source, the region of subcritical instability increases with increasing viscosity effect but reduces with increasing vertical permeability λ.

Resonant decay of kinetic Alfvén waves and implication on spectral cascading

A general equation describing the resonant nonlinear mode-coupling among kinetic Alfvén waves (KAWs) is derived using nonlinear gyrokinetic theory, which can be applied to study the potentially strong spectral energy transfer of KAWs. As a first application, the parametric decay of a pump KAW into two sideband KAWs is studied, with particular emphasis on cascading in the perpendicular wavenumber. It is found that, for the “co-propagating” cases with all three KAWs propagating in the same direction along the equilibrium magnetic field line, it exhibits a dual cascading character in the perpendicular wavenumber space; while for the “counter-propagating” cases with one sideband propagating in the opposite direction with respect to the pump wave, it instead, can exhibit both dual and inverse cascading behaviors. Furthermore, the implications of these findings on shear Alfvén waves regarding instability, nonlinear saturation, and charged particle transport in fusion plasmas are also discussed.

Nonlinear theory of the modulational instability at the ion–ion hybrid frequency and collapse of ion–ion hybrid waves in two-ion plasmas

We study the dynamics of two-dimensional (2D) nonlinear ion–ion hybrid waves propagating perpendicular to an external magnetic field in plasmas with two ion species. We derive nonlinear equations for the envelope of electrostatic potential at the ion–ion hybrid frequency to describe the interaction of ion–ion hybrid waves with low-frequency acoustic-type disturbances. The resulting nonlinear equations also take into account the contribution of the second harmonics of the ion–ion hybrid frequency. A nonlinear dispersion relation is obtained, and, for a number of particular cases, modulational instability growth rates are found. By neglecting the contribution of second harmonics, the phenomenon of the collapse of ion–ion hybrid waves is predicted. It is shown that taking into account the interaction with the second harmonic results in the existence of a stable two-dimensional soliton.

Observational constraints on extended Proca-Nuevo gravity and cosmology

Abstract We confront massive Proca-Nuevo gravity with cosmological observations. The former is a non-linear theory involving a massive spin-1 field, that can be extended incorporating operators of the Generalized Proca class, and when coupled to gravity it can be covariantized in a way that exhibits consistent and ghost-free cosmological solutions, without experiencing instabilities and superluminalities at the perturbative level. When applied at a cosmological framework it induces extra terms in the Friedmann equations, however due to the special non-linear construction the field is eliminated in favor of the Hubble function. Thus, the resulting effective dark energy sector is dynamical, however it contains the same number of free parameters with the ΛCDM concordance model. We use data from Supernovae Ia (SNIa) and Cosmic Chronometers (CC) observations and we construct the corresponding likelihood-contours for the free parameters. Interestingly enough, application of various information criteria, such as AIC, BIC and DIC, shows that the scenario of massive Proca-Nuevo gravity, although having exactly the same number of free parameters with ΛCDM paradigm, it is more efficient in fitting the data. Finally, the reconstructed dark-energy equation-of-state parameter shows statistical compatibility with the model-independent, data-driven reconstructed one.