Spectral Stability Analysis Research Articles

Study on Enhancing the Quantitative Performance of LIBS for Cu Element in Liquids Based on the Chitosan-Parafilm Enrichment Method

Abstract To enhance the accuracy of laser-induced breakdown spectroscopy (LIBS) in detecting heavy metal elements in solutions, a Chitosan (CS)-Parafilm (PM) enrichment method is proposed. This method involves drying chitosan-heavy metal complexes on a Parafilm substrate. During the drying of droplets, electrostatic attraction and chelation by CS effectively concentrate analytes. Additionally, the hydrophobic effect of the PM substrate induces Marangoni flow, which drags analytes from the bottom edge to the central top of the droplet surface, effectively suppressing the coffee ring effect (CRE).LIBS spectra were collected in a uniform array on the surface of sediments, and spectral stability was assessed through cumulative excitation spectra measurements. The results showed that the spectral stability of emission lines Cu I 324.754 nm and Cu I 327.396 nm decreased to 3.85% and 3.78%, respectively, indicating that the CS-PM enrichment method allows for uniform deposition of analytes within samples, effectively enhancing detection repeatability.Quantitative analysis of Cu elements using the CS-PM enrichment method was conducted using PSO-SVM, PSO-BPNN, and RF algorithms. The RF algorithm demonstrated the best predictive performance with Rp² of 0.979, RMSEP of 4.60 mg/L, MAE of 3.19 mg/L, and RPD of 6.75. These results indicate that CS-PM effectively improves spectral stability and quantitative analysis accuracy, providing a technical reference for enhancing the stability and quantitative performance of element analysis in liquids using LIBS.

Spectral Impedance-based Probabilistic Wideband Oscillatory Stability Analysis and Visualization

Spectral Impedance-based Probabilistic Wideband Oscillatory Stability Analysis and Visualization

Spectral stability analysis of the Dirichlet-to-Neumann map for fractional diffusion equations with a reaction coefficient

Spectral stability analysis of the Dirichlet-to-Neumann map for fractional diffusion equations with a reaction coefficient

Uncovering the Origins of Instability in Dynamical Systems: How Can the Attention Mechanism Help?

The behavior of the network and its stability are governed by both dynamics of the individual nodes, as well as their topological interconnections. The attention mechanism as an integral part of neural network models was initially designed for natural language processing (NLP) and, so far, has shown excellent performance in combining the dynamics of individual nodes and the coupling strengths between them within a network. Despite the undoubted impact of the attention mechanism, it is not yet clear why some nodes of a network obtain higher attention weights. To come up with more explainable solutions, we tried to look at the problem from a stability perspective. Based on stability theory, negative connections in a network can create feedback loops or other complex structures by allowing information to flow in the opposite direction. These structures play a critical role in the dynamics of a complex system and can contribute to abnormal synchronization, amplification, or suppression. We hypothesized that those nodes that are involved in organizing such structures could push the entire network into instability modes and therefore need more attention during analysis. To test this hypothesis, the attention mechanism, along with spectral and topological stability analyses, was performed on a real-world numerical problem, i.e., a linear Multi-Input Multi-Output state-space model of a piezoelectric tube actuator. The findings of our study suggest that the attention should be directed toward the collective behavior of imbalanced structures and polarity-driven structural instabilities within the network. The results demonstrated that the nodes receiving more attention cause more instability in the system. Our study provides a proof of concept to understand why perturbing some nodes of a network may cause dramatic changes in the network dynamics.

The stability of the b-family of peakon equations

In the present work we revisit the b-family model of peakon equations, containing as special cases the b = 2 (Camassa–Holm) and b = 3 (Degasperis–Procesi) integrable examples. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for b < 1. We explore numerically these ideas in the realm of fixed-point iterations, spectral stability analysis and time-stepping of the model for the different parameter regimes. In particular, we identify exact, stationary (spectrally stable) lefton solutions for , and for , we dynamically identify ramp-cliff solutions as dominant states in this regime. We complement our analysis by examining the breakup of smooth initial data into stable peakons for b > 1. While many of the above dynamical features had been explored in earlier studies, in the present work, we supplement them, wherever possible, with spectral stability computations.

On the stability of solitary wave solutions for a generalized fractional Benjamin–Bona–Mahony equation

In this paper we establish a rigorous spectral stability analysis for solitary waves associated to a generalized fractional Benjamin–Bona–Mahony type equation. Besides the well known smooth and positive solitary wave with large wave speed, we present the existence of smooth negative solitary waves having small wave speed. The spectral stability is then determined by analysing the behaviour of the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum at the solitary wave. Since the analytical solution is not known, we generate the negative solitary waves numerically by using Petviashvili method. We also present some numerical experiments to observe the stability properties of solitary waves for various values of the order of nonlinearity and fractional derivative. Some remarks concerning the orbital stability are also celebrated.

An image auxiliary method for laser-induced breakdown spectroscopy analysis of coal particle flow

An image information-assisted spectral screening method was proposed for improving the coal particle flow detection performance via LIBS and its positive effects on spectral stability and quantitative analysis were verified.

Dark–dark soliton breathing patterns in multi-component Bose–Einstein condensates

In this work, we explore systematically various SO(2)-rotation-induced multiple dark–dark (DD) soliton breathing patterns obtained from stationary and spectrally stable multiple dark–bright (DB) and DD waveforms in trapped one-dimensional, two-component atomic Bose–Einstein condensates. The stationary states stemming from the associated linear limits (as the eigenfunctions of the quantum harmonic oscillator problem) are parametrically continued to the nonlinear regimes by varying the respective chemical potentials, i.e. from the low-density linear limits to the high-density Thomas–Fermi (TF) regimes. We perform a Bogolyubov–de Gennes spectral stability analysis to identify stable parametric regimes of these states, finding a wide range of stability intervals in the TF regimes for all of the states considered herein. Upon applying an SO(2)-rotation to stable steady states, one-, two-, three-, four-, and many DD soliton breathing patterns are observed in the numerical simulations. Furthermore, analytic solutions up to three DB solitons in the homogeneous setting, and three-component systems are also investigated.

Stability of topological edge states under strong nonlinear effects

We examine the role of strong nonlinearity on the topologically-robust edge state in a one-dimensional system. We consider a chain inspired from the Su-Schrieffer-Heeger model, but with a finite-frequency edge state and the dynamics governed by second-order differential equations. We introduce a cubic onsite-nonlinearity and study this nonlinear effect on the edge state's frequency and linear stability. Nonlinear continuation reveals that the edge state loses its typical shape enforced by the chiral symmetry and becomes generally unstable due to various types of instabilities that we analyze using a combination of spectral stability and Krein signature analysis. This results in an initially-excited nonlinear-edge state shedding its energy into the bulk over a long time. However, the stability trends differ both qualitatively and quantitatively when softening and stiffening types of nonlinearity are considered. In the latter, we find a frequency regime where nonlinear edge states can be linearly stable. This enables high-amplitude edge states to remain spatially localized without shedding their energy, a feature that we have confirmed via long-time dynamical simulations. Finally, we examine the robustness of frequency and stability of nonlinear edge states against disorder, and find that those are more robust under a chiral disorder compared to a non-chiral disorder. Moreover, the frequency-regime where high-amplitude edge states were found to be linearly stable remains intact in the presence of small amount of disorder of both types.

Bifurcation analysis of stationary solutions of two-dimensional coupled Gross–Pitaevskii equations using deflated continuation

Recently, a novel bifurcation technique known as deflated continuation was applied to the single-component nonlinear Schrödinger (NLS) equation with a parabolic trap in two spatial dimensions. This bifurcation analysis revealed previously unknown solutions, shedding light on this fundamental problem in the physics of ultracold atoms. In the present work, we take this a step further by applying deflated continuation to two coupled NLS equations, which – feature a considerably more complex landscape of solutions. Upon identifying branches of solutions, we construct the relevant bifurcation diagrams and perform spectral stability analysis to identify parametric regimes of stability and instability and to understand the mechanisms by which these branches emerge. The method reveals a remarkable wealth of solutions. These include both well-known states arising from the Cartesian and polar small amplitude limits of the underlying linear problem, but also a significant number of more complex states that arise through (typically pitchfork) bifurcations.

The effect of sample surface roughness on the microanalysis of microchip laser-induced breakdown spectroscopy

In the microanalysis of laser-induced breakdown spectroscopy, the influence of surface roughness on spectral stability and quantitative analysis capability was studied for the first time when the laser ablation crater diameter was approximately 10 μm.

Ring dark solitons in three-dimensional Bose-Einstein condensates

In this work we present a systematic study of the three-dimensional extension of the ring dark soliton examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately, but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semi-analytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov-de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.

Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates.

We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.

Enhancement in heat and mass transfer over a permeable sheet with Newtonian heating effects on nanofluid: Multiple solutions using spectral method and stability analysis

This paper investigates the Newtonian heating effect on nanofluid flow over a nonlinear permeable stretching / shrinking sheet near the region of stagnation point. Only two important mechanisms on the transportation of nanoparticles in base fluid are discussed: the Brownian motion and thermophoresis. This physical problem is modelled using the Buongiorno (ASME J. Heat Transfer 128, 240 (2006) model in terms of nonlinear governing partial differential equations and transformed into dimensionless ordinary differential equations by using similarity transformation and the solution is calculated using the numerical scheme known as the Chebyshev spectral collocation method. The main interest of this study is the region of the boundary layer where viscous effects are dominant. Dual solutions are reported against the shrinking parameter in which the first solution is stable due to positive eigenvalues and the second is unstable due to negative eigenvalues and ranges of these solutions are effected by the suction parameter which is discussed using graphs and tables. The effects of dimensionless parameters, namely, velocity ratio, suction, Schmidt number, Prandtl number, thermophoresis and Brownian motion on temperature and concentration profiles, skin friction coefficient and Nusselt number are also shown using graphs. For the validity of the applied scheme, a comparison is established with published studies in the limiting case. Through the results, it is concluded that temperature and concentration increase by increasing the values of the thermophoresis parameter and the opposite behaviour is observed in the case of Brownian motion and Schmidt number. Skin friction coefficient, Nusselt and Sherwood numbers increase on increasing the suction parameter. Also, an enhancement in temperature and concentration profiles is observed in the presence of Newtonian heating parameter.

On spatio-temporal dynamics of sine-Gordon soliton in nonlinear non-homogeneous media using fully implicit spectral element scheme

One- and two-dimensional sine-Gordon equation in non-homogeneous media is considered. Sine-Gordon equation exhibits soliton-like solution whose existence and behaviour in non-homogeneous media is studied. Various non-homogeneous media are discussed which include nonlinear smooth as well as discontinuous density variations. Time-dependent continuous and discontinuous density variations which can model geodynamical processes like lithospheric deformation, fault dynamics, etc. are also discussed. In one dimension, dynamics of kink which are topological soliton is studied under such density variations, whereas in two dimensions, circular-ring soliton dynamics has been scrutinized. The governing sine-Gordon equation is solved numerically using higher order Legendre polynomial-based spectral element method. Spectral stability analysis of the numerical scheme shows the strong dependence of a critical time step not only on density value but also on the nature of the distribution. It is shown that for smooth density variation the critical time step reduces drastically which makes explicit scheme very expensive. To reduce the computational cost, unconditionally stable fully implicit spectral element scheme is proposed which is also energy conservative.

Stability of Vertically Traveling, Pre-tensioned, Heavy Cables

We study the stability of a pre-tensioned, heavy cable traveling vertically against gravity at a constant speed. The cable is modeled as a slender beam incorporating rotary inertia. Gravity modifies the tension along the traveling cable and introduces spatially varying coefficients in the equation of motion, thereby precluding an analytical solution. The onset of instability is determined by employing both the Galerkin method with sine modes and finite element (FE) analysis to compute the eigenvalues associated with the governing equation of motion. A spectral stability analysis is necessary for traveling cables where an energy stability analysis is not comprehensive, because of the presence of gyroscopic terms in the governing equation. Consistency of the solution is checked by direct time integration of the governing equation of motion with specified initial conditions. In the stable regime of operations, the rate of change of total energy of the system is found to oscillate with bounded amplitude indicating that the system, although stable, is nonconservative. A comprehensive stability analysis is carried out in the parameter space of traveling speed, pre-tension, bending rigidity, external damping, and the slenderness ratio of the cable. We conclude that pre-tension, bending rigidity, external damping, and slenderness ratio enhance the stability of the traveling cable while gravity destabilizes the cable.

Higher order scheme for two-dimensional inhomogeneous sine-Gordon equation with impulsive forcing

In this paper higher order scheme is presented for two-dimensional sine-Gordon equation. Higher order Legendre spectral element method is used for space discretization which is basically a domain decomposition method and it retains all the advantages of spectral and finite element methods. Spectral stability analysis is performed for both homogeneous and inhomogeneous sine-Gordon equations which gives implicit expressions of critical time step. Various test cases are solved which shows the robustness and accuracy of the proposed scheme. Moreover, experimental order of convergence is obtained which gives optimal convergence rate. It is also shown that the proposed scheme exhibit conservation of energy for undamped sine-Gordon equation. In realistic scenario defects are predominant in nature. sine-Gordon equation with defect can be effectively modeled by additional impulsive forcing term. In the second part of this paper, proposed higher order scheme is used to solve inhomogeneous sine-Gordon equation with impulsive loading. Both constant as well as time-dependent strength of impulsive forcing are used. Soliton solution to sine-Gordon equation is analyzed under the action of such forcing. Different Dirac delta representations are discussed which can accurately replicate behavior of impulsive loading. This is important because, dynamics of soliton behavior strongly depends on the impulsive force representation. Various conclusions are made based on this study.

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.

Vortex precession dynamics in general radially symmetric potential traps in two-dimensional atomic Bose-Einstein condensates

We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency $\omega$ on the chemical potential $\mu$, this is no longer true for a general potential $V(r) \propto r^p$. Our calculations suggest that for $p>2$, the precession frequency scales with $\mu$ as $\omega \sim \mu^{-2/p}$. This theoretical prediction is corroborated by numerical computations, both at the level of spectral (Bogolyubov-de Gennes) stability analysis by identifying the relevant precession mode dependence on $\mu$, but also through direct numerical computations of the vortex evolution in the large $\mu$, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the radius of an initially displaced from the center vortex is examined and the corresponding predictions are tested against numerical results.

Floquet analysis of Kuznetsov-Ma breathers: A path towards spectral stability of rogue waves.

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a nontrivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov-Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.