Quadratic Nonlinearity Research Articles

On the absence of weak solutions for a sequential time-fractional Klein-Gordon equation with quadratic nonlinearity

On the absence of weak solutions for a sequential time-fractional Klein-Gordon equation with quadratic nonlinearity

Heralded pure single-photon sources using nanophotonic waveguides with quadratic and cubic nonlinearities

This paper presents, to our knowledge, a new approach in developing integrated pure heralded single-photon sources based on the interplay between the spontaneous four-wave mixing and sum-frequency generation parametric processes. We introduce a comprehensive quantum model to exploit this interplay in AlGaAs and LiNbO3 nanophotonic waveguides. The developed model is used to assess the performance of the sources based on the photon-pair generation and the associated spectral purity. We find that this approach can remarkably improve the spectral purity of low-pure generated photon pairs, relaxing the restrictions on the structure design and the used pump wavelength. In addition, it overcomes the current hurdles in implementing on-chip photon detectors operating at room temperature, paving the way for advanced applications in integrated quantum photonics and information processing.

A unified framework for the analysis of accuracy and stability of a class of approximate Gaussian filters for the Navier–Stokes equations

Abstract Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. In these cases, nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation (DA) techniques. A number of DA algorithms used commonly in practice, such as the Ensemble Kalman Filter (EnKF) or Ensemble Square Root Kalman Filter (EnSRKF), suffer from serious drawbacks such as catastrophic filter divergence and filter instability. The analysis of stability and accuracy of these DA schemes has thus far focused either on finite-dimensional dynamics, or on complete observations in case of infinite-dimensional dissipative systems. We develop a unified framework for the analysis of several well-known and empirically efficient DA techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to DA. We consider the ‘model’ dynamics governed by the two-dimensional incompressible Navier–Stokes equations (NSEs) and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time DA techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced NSEs. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the L 2 norm and of the H 1 Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the DA algorithms and covariance inflation and localization which are employed in practice for improved filter performance.

On Laplace Invariants of a Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities

On Laplace Invariants of a Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities

Decomposition of nonlinear collision operator in quantum Lattice Boltzmann algorithm

We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.

Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

Quadratic-soliton-enhanced mid-IR molecular sensing

Optical solitons have long been of interest both from a fundamental perspective and because of their application potential. Both cubic (Kerr) and quadratic nonlinearities can lead to soliton formation, but quadratic solitons can practically benefit from stronger nonlinearity and achieve substantial wavelength conversion. However, despite their rich physics, quadratic cavity solitons have been used only for broadband frequency comb generation, especially in the mid-infrared. Here, we show that the formation dynamics of mid-infrared quadratic cavity solitons, specifically temporal simultons in optical parametric oscillators, can be effectively leveraged to enhance molecular sensing. We demonstrate significant sensitivity enhancement while circumventing constraints of traditional cavity enhancement mechanisms. We perform experiments sensing CO2 using cavity simultons around 4 μm and achieve an enhancement of 6000. Additionally, we demonstrate large sensitivity at high concentrations of CO2, beyond what can be achieved using an equivalent high-finesse linear cavity by orders of magnitude. Our results highlight a path for utilizing quadratic cavity nonlinear dynamics and solitons for molecular sensing beyond what can be achieved using linear methods.

Vortex solitons in rotating quasi-phase-matched photonic crystals.

We present an approach to generate stable vortex solitons (VSs) in rotating quasi-phase-matched photonic crystals with quadratic nonlinearity. The photonic crystal is introduced with a checkerboard structure, which can be realized using available technology. The VSs are constructed as four-peak vortex modes of two types: rhombuses and squares. Control parameters, including the power, rotating frequency, and size of each square cell, affect the distribution and stability range of these VSs. The tightly binding rhombic VSs realize the system's ground state, which features the lowest value of the Hamiltonian. By introducing rotation, stable VSs with topological charges l = ±1 and ±2 are observed, and the VSs turn from a quadrupole to a vortex-like state. The generation and modulation of stable VSs in rotating quasi-phase-matched photonic crystals demonstrate promising applications in optical communication systems, optical tweezers, and quantum information processing, where precise control over light propagation and vortex states is crucial.

Experimental analysis of Local Defect Resonance in an aluminum plate

In the context of non-destructive evaluation and testing of structures, the use of Local Defect Resonance is promising for the detection of delamination-type defects in composites. In this context, an experimental analysis Local Defect Resonance is proposed. It is here carried out on a model system: a thin aluminum plate with a Flat Bottom Hole (FBH) defect consisting in a self-adhesive aluminum sheet (a shim) adhesively bonded over a circular through hole. First, the local defect resonance frequencies and mode shapes are analyzed assuming linear vibrations of the structure. Then, a nonlinear approach is used to compare two types of FBH defects, a first flawless one and a second one including a damaged area. For this purpose a high amplitude vibration around the resonance frequencies are used in order to generate high order harmonics. This enhancement is identified as associated with mainly quadratic nonlinearity.

On the parametric few-cycle light bullets

Numerical simulation demonstrates that (2D+1) few-cycle (3–5 oscillations under the envelope) light bullets may form in the medium with quadratic nonlinearity and group velocity anomalous dispersion under conditions of second-harmonic generation. It is shown that as the number of oscillations under the envelope decreases, the parameters of such two-frequency solitons change.

Vortex light bullets in rotating Quasi-Phase-Matched photonic crystals

We present a methodology for the generation of stable vortex light bullets (LBs) in rotating Quasi-Phase-Matched (QPM) photonic crystals with quadratic nonlinearity. The photonic crystal is designed with a checkerboard structure, which is feasible to realize by using contemporary technological advancements. Within this framework, square- and rhombus-shaped LBs are observed, both of which are constructed as four-peak vortex mode. The control parameters include the effective phase mismatch, power, rotating frequency, and the size of checkerboard cells. These parameters play key roles in determining the distribution and stability domains of vortex LBs. In contrast to the stable vortex solitons observed in two-dimensional (2D) quadratic systems, the LBs investigated in the 3D rotating system exhibit narrower stability domains within the system’s parameter space. The rotating frequency results in the transition of LBs from quadrupole to traditional vortex modes. Potential applications of this research lie in the field of optical communications and information processing.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Numerical studies of one-way Lamb and SH mixing method in composite laminates with transverse-isotropic quadratic nonlinearity

Abstract The resonant behavior of one-way Lamb and SH (shear horizontal) mixing method in composite laminates with transverse-isotropic quadratic nonlinearity is investigated through numerical simulations in this paper. Different from previous studies, the composite constitutive model is combined from orthotropic elasticity and transverse-isotropic quadratic nonlinearity, which is implemented by ABAQUS/VUMAT subroutine. When two fundamental waves (S 0-mode Lamb waves and SH 0 waves) mix in composite laminates with quadratic nonlinearity, the resonant SH 0 waves can be generated with the resonance condition ω S 0 /ω SH 0 = 2 κ/(κ + 1). Meanwhile, the relationships between the acoustic nonlinear parameter (ANP) and damage degree, fundamental frequency, frequency deviation, propagation distance are also investigated. Moreover, the method of locating the damage region in composite laminates is proposed and verified by using the resonant wave time-domain signal.

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Abstract Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Effect of adiabatic trapping of electrons on the nonlinear evolution of ion temperature gradient driven drift mode in a dispersive plasma

We have theoretically investigated the effect of adiabatic trapping of electrons on ion temperature gradient (ITG) driven nonlinear drift mode in a warm and dispersive electron-ion plasma. For this purpose, we have incorporated the gradients in the background plasma density, ion temperature and ambient magnetic field and derived two different nonlinear partial differential equations (NLPDEs). One of them contains only fractional nonlinearity while the other one incorporates the effect of both quadratic and fractional nonlinearities. We have obtained the exact solutions of these NLPDEs by using the functional variable method. We have used the graphical analysis to carry out the parametric study of the obtained solutions for the Tokamak plasma parameters. We have shown that the amplitude and the width of these nonlinear structures depend on the plasma parameters like T e , T i and η i . This work may be helpful to understand the effect of electron trapping on the low frequency drift type modes in laboratory and space plasmas.

Novel stochastic embedded solitons with quadratic nonlinear susceptibility in the presence of multiplicative noise

This paper addresses the modeling of optical systems with stochastic quadratic nonlinearity for the first time, a novel and challenging research area within nonlinear optics. By incorporating multiplicative white noise and quadratic nonlinear susceptibility, the study presents an innovative approach to recovering optical solutions. Leveraging the G′G -expansion method and extended Kudryashov’s method, new stochastic exact solutions are derived, encompassing bright, dark, singular, and trigonometric solitons. Graphical representations aid in understanding these solutions’ characteristics. Insights into the stochastic nature of optical solutions under various conditions are provided, offering valuable contributions to nonlinear optics and potential applications in telecommunications and materials science.

Model reduction of dynamical systems with a novel data-driven approach: The RC-HAVOK algorithm.

This paper introduces a novel data-driven approximation method for the Koopman operator, called the RC-HAVOK algorithm. The RC-HAVOK algorithm combines Reservoir Computing (RC) and the Hankel Alternative View of Koopman (HAVOK) to reduce the size of the linear Koopman operator with a lower error rate. The accuracy and feasibility of the RC-HAVOK algorithm are assessed on Lorenz-like systems and dynamical systems with various nonlinearities, including the quadratic and cubic nonlinearities, hyperbolic tangent function, and piece-wise linear function. Implementation results reveal that the proposed model outperforms a range of other data-driven model identification algorithms, particularly when applied to commonly used Lorenz time series data.

Dynamic Analysis and PD Control in a 12-Pole Active Magnetic Bearing System

This paper conducts an in-depth study on the dynamic stability and complex vibration behavior of a 12-pole active magnetic bearing (AMB) system considering gravitational effects under a PD controller. Firstly, based on electromagnetic theory and Newton’s second law, a two-degree-of-freedom control equation of the system, including PD control terms and gravitational effects, is constructed. This equation involves not only parametric excitation, quadratic nonlinearity, and cubic nonlinearity but also a more pronounced coupling effect between the magnetic poles due to the presence of gravity. Secondly, using the multi-scale method, a four-dimensional averaged equation of the system in Cartesian and polar coordinates is derived. Finally, through numerical analysis, the system’s amplitude–frequency response, motion trajectory, the relationship between energy and amplitude, and global dynamic behaviors such as bifurcation and chaos are discussed in detail. The results show that the PD controller significantly affects the system’s spring hardening/softening characteristics, excitation, amplitude, energy, and stability. Specifically, increasing the proportional gain can quickly suppress the rotor’s motion, but it also increases the system’s instability. Adjusting the differential gain can transition the system from a chaotic state to a stable periodic motion.

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

First transition dynamics of reaction–diffusion equations with higher order nonlinearity

AbstractIn this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of and . The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section.