Linear Stability Research Articles

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

The Effect of Heavy Inclusions in the Upper Crust on Thermoconvective Instability of the Lithosphere

The method of linear theory for small perturbations is used to study the stability of heavy inclusions in the upper crust. The existence of such inclusions of increased density is associated with chemical heterogeneity or phase transitions. At small deformations considered in the linear stability theory, the material undergoes transient creep which is described by the Andrade rheological model. Under this rheology, the convective instability of thickened segments of the lithosphere is oscillatory: the increment is a complex number whose real part is close to zero (threshold instability). Heavy inclusions in the upper crust introduce a small complex correction to the increment that determines the convective instability of the lithosphere, slightly increasing the real and imaginary parts of the increment, enhancing the instability and increasing the oscillation frequency.

Throughflow effect on bi-disperse convection in Rivlin-Ericksen fluid

In this investigation, we delve into the influence of throughflow on the phenomenon of bi-disperse convection within Rivlin-Ericksen fluid. In the context of examining bi-disperse convection within this specific type of fluid, the throughflow effect is considered to have a uniform vertical distribution. The primary focus of this study centers on evaluating the system’s linear stability. To achieve this, we employ normal mode analysis to compute the Darcy-Rayleigh number at the onset of convection. This Darcy-Rayleigh number is computed for both stationary and oscillatory convection modes. Furthermore, we conduct a comprehensive analysis and present the results in graphical form to illustrate the impact of various parameters, including Peclet number and kinematic viscoelastic parameter, on both stationary and oscillatory convection. Our research findings demonstrate that when Peclet number Pr1 < 0, it leads to destabilising effect on both stationary and oscillatory convections. Conversely, when Peclet number Pr1 > 0, it induces stabilising effect on both stationary as well as oscillatory convections.

Development of a neural network model for peeling–ballooning stability analysis in the KSTAR tokamak pedestals

The neural network model, MISHKA-NN is developed to mitigate the computational burden associated with the linear ideal magnetohydrodynamic (MHD) stability analysis of the pedestal based on the peeling–ballooning (P–B) model. By utilizing both 1D plasma profiles (current density, pressure gradient, and safety factor) and 0D parameters (plasma geometry, total current, and toroidal mode number), the model predicts linear growth rate of edge-localized ideal MHD instability in a given equilibrium state. By enabling the prediction of each instability within a second, the model reduces the time required for plotting a pedestal P–B stability diagram ( j−α diagram) from approximately 100 CPU hours to a few CPU minutes. Notably, even with the utilization of parametric pressure and current profiles and plasma boundary shapes for the training dataset, the model shows a satisfactory level of performance in benchmarking the j−α diagram for the reconstructed equilibrium from a KSTAR tokamak experiment. We anticipate the model to serve as a versatile alternative to 2D linear MHD stability codes, alleviating numerical costs.

Rotating Stall Inception Prediction Using an Eigenvalue-Based Global Instability Analysis Method

The accurate prediction of rotating stall inception is critical for determining the stable operating regime of a compressor. Among the two widely accepted pathways to stall, namely, modal and spike, the former is plausibly believed to originate from a global linear instability, and experiments have partially confirmed it. As for the latter, recent computational and experimental findings have shown it to exhibit itself as a rapidly amplified flow perturbation. However, rigorous analysis has yet to be performed to prove that this is due to global linear instability. In this work, an eigenanalysis approach is used to investigate the rotating stall inception of a transonic annular cascade. Steady analyses were performed to compute the performance characteristics at a given rotational speed. A numerical stall boundary was first estimated based on the residual convergence behavior of the steady solver. Eigenanalyses were then performed for flow solutions at a few near-stall points to determine their global linear stability. Once the relevant unstable modes were identified according to the signs of real parts of eigenvalues, they were examined in detail to understand the flow destabilizing mechanism. Furthermore, time-accurate unsteady simulations were performed to verify the obtained eigenvalues and eigenvectors. The eigenanalysis results reveal that at the rotating stall inception condition, multiple unstable modes appear almost simultaneously with a leading mode that grows most rapidly. In addition, it was found that the unstable modes are continuous in their nodal diameters, and are members of a particular family of modes typical of a dynamic system with cyclic symmetries. This is the first time such an interesting structure of the unstable modes is found numerically, which to some extent explains the rich and complex results constantly observed from experiments but have never been consistently explained. The verified eigenanalysis method can be used to predict the onset of a rotating stall with a CPU time cost orders of magnitude lower than time-accurate simulations, thus making compressor stall onset prediction based on the global linear instability approach feasible in engineering practice.

Linear temporal stability of Jeffery–Hamel flow of nanofluids

Flow stability plays a key role in transition to turbulence in various systems. This transition initiates with disturbances appearing in the laminar base flow, potentially amplifying over time based on flow and fluid parameters. In response to these amplified disturbances, the flow undergoes successive stages of different laminar flows, ultimately transitioning to turbulence. One influential parameter affecting flow stability is the nanoparticle volume fraction (ϕ) in nanofluids, extensively employed in thermofluid systems like cooling devices to enhance fluid thermal conductivity and the heat transfer coefficient. Focusing on the impact of nanoparticles on Jeffery–Hamel flow stability, this study assumes fluid properties are temperature- and pressure-independent, exclusively examining the momentum transfer aspect. The analysis commences by deriving the base laminar flow solution. Subsequently, linear temporal stability analysis is employed, imposing infinitesimally-small perturbations on the base flow as a modified form of normal modes. A generalized Orr–Sommerfeld equation is derived and solved using a spectral method. Results indicate that, assuming nanofluid viscosity as μnf=μf/(1−ϕ)2.5, nanoparticle effects on momentum transfer and flow stability hinge on the ratio of nano-solid particle density to base fluid density (Rρ=ρs/ρf). For ϕ∈(0,0.1], flow stabilization occurs with ϕ when Rρ<3.5000, while destabilization is observed when Rρ>4.0135. Notably, nanoparticles exhibit a negligible impact on flow stability when 3.5000≤Rρ≤4.0135.

Macroscopic trends of linear tearing stability in cylindrical current profiles

The likelihood of realising tokamak power-plants will be greatly improved by the discovery of high-gain equilibria that resist the formation of small islands and hence avoid the disruptive neoclassical tearing mode. We propose a series of studies to understand how simple tokamak design can leverage aspects of tearing onset physics to maximise passive resistance to island formation. Here we investigate the variation that current profiles can bring about in preventing tearing onset through the cylindrical linear tearing stability parameter Δ′ . A database of 159148 realistic pilot-plant current profiles was generated with Monte Carlo sampling, and the distribution of Δ′ values was linked with interpretable profile characteristics. In agreement with prior theoretical and experimental studies, Δ′ was found to be strongly correlated with the existence and steepness of a local toroidal current well or hill, with the former destabilising and the latter stabilising. In the absence of these two cases, the remaining Δ′ values were linearly bounded by the toroidal current gradient at the rational surface.

Generalized naming game and Bayesian naming game as dynamical systems.

We study the β model (β-NG) and the Bayesian Naming Game (BNG) as dynamical systems. By applying linear stability analysis to the dynamical system associated with the β model, we demonstrate the existence of a nongeneric bifurcation with a bifurcation point β_{c}=1/3. As β passes through β_{c}, the stability of isolated fixed points changes, giving rise to a one-dimensional manifold of fixed points. Notably, this attracting invariant manifold forms an arc of an ellipse. In the context of the BNG, we propose modeling the Bayesian learning probabilities p_{A} and p_{B} as logistic functions. This modeling approach allows us to establish the existence of fixed points without relying on the overly strong assumption that p_{A}=p_{B}=p, where p is a constant.

Stability of asymmetric Schwarzschild–Rindler–de Sitter thin shell wormhole

The paper examines the dynamics of asymmetric thin shell wormholes that connect two distinct spacetimes using the cut and paste technique. The focus is on analyzing the linear stability of these wormholes by considering radial perturbations and utilizing the modified generalized Chaplygin gas equation of state. The specific case of an asymmetric wormhole connecting Schwarzschild–Rindler spacetime to Schwarzschild–Rindler–de Sitter space–time is analyzed using this formalism. Our investigation uncovers the existence of both stable and unstable regions, which are contingent upon the appropriate selection of various parameters within the metric spacetime and equation of state. Additionally, we determine that stability regions exist as a consequence of the square speed of sound. By increasing the value of the cosmological constant, the stability region is expanded. Furthermore, the stability regions are augmented by the influence of Rindler parameters, while the stability regions are also affected by adjustments in the equation of state parameters, leading to their enlargement.

Some optical solitons and modulation instability analysis of (3 + 1)-dimensional nonlinear Schrödinger and coupled nonlinear Helmholtz equations

In this study, we acquired some optical solitons of (3 + 1) dimensional nonlinear Schrödinger equation (3DNLSE) and coupled nonlinear Helmholtz equations (CNLHE) by using generalized G′G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(\\frac{{G}^{\\prime}}{G}\\right)$$\\end{document}-expansion method (GEM). We are able to derive exponential, trigonometric, and hyperbolic solutions. We notice that Mathematica 11.2 offer the equations for these answers. Aside from that, we display some of the solution’s graphic performance. Recently, accurate traveling-wave solutions for nonlinear partial differential equations (NLPDEs) have been acquired using this technique. Through linear stability analysis, we derive an analytical expression for the instability gain and analyze its main characteristics. Finally, we assess the robustness and stability of the solitary waves through numerical simulations and compare the accuracy of the analytical and numerical results. The obtained ultra-short optical solitons can be widely used as a guide for practical applications in telecommunications domain.

Gravitating kinks with asymptotically flat metrics

In this work, we consider a two-dimensional (2D) dilaton gravity model where the dilaton kinetic term is modified by an additional derivative coupling term . In the case with a canonical scalar matter field, the field equations of this model have a simple first-order formalism, from which exact static kink solutions can be constructed. The novelty of these solutions is that the corresponding metric can be asymptotically flat rather than asymptotically anti-de Sitter. The linear stability and the localization of scalar matter fields are also studied. It was found that the solutions are stable against small linear perturbations, and the localization of scalar matter fields can be realized by introducing scalar-kink interactions.

Stability of fluid flow in a porous medium with uniform cross-flow and velocity slip

This work examines the evolution of small perturbations in a plane Poiseuille flow through a porous medium bounded by porous walls implemented with the Brinkman flow model. The effect of uniform cross-flow across the channel along with slip imposed on the channel walls is examined for the associated fluid flow. The modified Orr–Sommerfeld equation for the flow system under consideration is derived and then solved numerically using Chebyshev spectral collocation method. The acquired sufficient conditions ensure that fluid motion will remain stable even when the flow system is widely perturbed. The linear stability characteristics and the onset of instability is discussed for a wide range of cross-flow Reynolds number, slip length and porosity parameter. The above mentioned flow parameters have a significant stabilizing effect on the fluid flow that is depicted with neutral stability curves, growth rate dispersion curves and eigenspectrums. Asymmetric slip allows the critical Reynolds number to increase rapidly when the upper channel wall experiences more slip than the lower.

Hydromagnetic Thermosolutal Instability of an Oldroyd Nanofluid in a Porous Medium

The onset of double-diffusive convection of an Oldroyd nanofluid fluid saturated in a porous medium, heated and soluted from below, is examined both analytically and numerically under the linear stability analysis in the presence of a horizontal magnetic field and physically realistic boundary condition on the volume fraction of nanoparticles. The resulting eigenvalue problem is solved numerically by employing Galerkin method. We recover the important works of Umavathi and Prathap Kumar (J. C. Umavathi and J. Prathap Kumar, J. Heat Transfer 139, 012401 (2017)), Jaimala et al. (R. Jaimala, Singh, and V. K. Tyagi, Int. J. Heat Mass Transfer 111, 451 (2017); Jaimala, R. Singh, and V. K. Tyagi, Int. J. Heat Mass Transfer 125, 290 (2018)) and Kuznetsov and Nield (A. V. Kuznetsov and D. A. Nield, International Journal of Thermal Sciences 77, 126 (2014)) as special cases. The important findings include: (i) the independence of thermal Rayleigh-Darcy number for stationary convection upon relaxation and retardation parameters, (ii) the parameters Rn, Rs, Le, NA and λ1 (absent in case of stationary convection) enhance both the stationary and oscillatory convections and the parameter Q, ε and λ2 (absent in case of stationary convection) delay the occurrence of both convections, (iii) instability first sets in as oscillatory convection and (iv) a number of sufficient conditions for the existence and also for the non-existence of oscillatory convection are obtained analytically but only stated (without proof).

Local and extensive fluctuations in sparsely interacting ecological communities.

Ecological communities with many species can be classified into dynamical phases. In systems with all-to-all interactions, a phase where species abundances always reach a fixed point and a phase where they continuously fluctuate have been found. The dynamics when interactions are sparse, with each species interacting with only a few others, has remained largely unexplored. Here we study a system of sparse interactions, first when interactions are of constant strength and completely unidirectional, and then when adding variability and bidirectionality. We show that in this case a phase unique to the sparse setting appears in the phase diagram, where for the same control parameters different communities may reach either a fixed point or a state where the abundances of only a finite subset of species fluctuate, and we calculate the probability for each outcome. These fluctuating species are organized around short cycles in the interaction graph, and their abundances undergo large nonlinear fluctuations. We characterize the approach from this phase to a phase with extensively many fluctuating species, and show that the probability of fluctuations grows continuously to one as the transition is approached, and that the number of fluctuating species diverges. This is qualitatively distinct from the transition to extensive fluctuations coming from a fixed point phase, which is marked by a loss of linear stability. The differences are traced back to the emergent binary character of the dynamics when far from short cycles.

Emergence of non-trivial solutions from trivial solutions in reaction–diffusion equations for pattern formation

Reaction–diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction–diffusion models: the Schnakenberg model and the Gray–Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

Nonlinear stability of two superimposed electrified dusty fluids of type Rivlin-Ericksen: Non-perturbative approach

The nonlinear instability of a planar interface between two overlapping Rivlin-Ericksen viscoelastic liquids, in the occurrence of fine dust and vertical electric fields throughout permeable media, and the influence of surface tension is studied. The prototype structure is assumed to be a two-dimensional groundwork for the sake of simplicity. The growing approval of several disciplines of practical physics and technology serves as motivation for analyzing this problem. Using viscous potential theory, the mathematical procedure is condensed. To properly control electro convection in liquid crystals, which is widely used in various displays technologies, it was crucial to have a thorough comprehension of stability. Gaining understanding of the behavior of viscoelastic fluids at interfaces was crucial in polymer manufacturing industries. Employing the linear equations of movement with the suitable nonlinear bounder circumstances forms the basis of the considered nonlinear technique. A nonlinear partial differential equation that evaluates the interface movement is the objective of the process. Since linear stability has frequently been investigated, the present situation will be constructed only in the nonlinear sense, where a number of dimensionless physical numerals are achieved. The non-perturbative methodology is used to accomplish the nonlinear standards. The main idea behind the novel performance is to convert an ordinary nonlinear ordinary differential equation into a linear one. The new technique differs from the conventional perturbation methods in that it may be used to exactly and correctly analyze the behavior of the strong nonlinear aspects of the interface displacement. This innovative method is established by using He's frequency formula. Using the nonlinear distinguishing equation, the special situation of the real and the complex coefficients is accomplished. It was observed that the instability zone expands as the electric field, porosity of porous medium, and density ratio of two fluids rise. The stability configuration is enhanced by the viscoelasticity parameter, Bond number, and ratio of kinematic viscoelasticity.

Establishment of Linear Stability Analysis Method of Hypersonic Boundary Layer Reflecting Porous Surface Effect

Establishment of Linear Stability Analysis Method of Hypersonic Boundary Layer Reflecting Porous Surface Effect

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Thermal response of nematicons in a parabolic potential

The thermal response of nematicons in a parabolic potential has been numerically studied. Single-peak nematicons exist only in the absence of thermal response coefficients. Because focusing reorientational nonlinearity is dominant in this case. In the presence of thermal response, the competition between focusing reorientational and defocusing thermal nonlinearities leads to the transformation of single-peak to double-peak nematicons. In this domain, the defocusing thermal nonlinearity is greater than the focusing reorientational nonlinearity, resulting in double-peak nematicons. The energy landscape experienced by the light beam within the medium is modified by the competing nonlinearities. The presence of both focusing reorientational and defocusing thermal nonlinearities creates multiple maxima in the energy landscape, allowing for the stabilization of double-peak nematicons as equilibrium states. When a parabolic potential is present, periodic oscillations can be obtained in nematicon. For small values of thermal response coefficients, double-peak nematicons having periodic oscillations are obtained. The thermal response coefficients have significant impacts on the wavelength of the oscillations of double-peak nematicon. The wavelength has been found to increase with increasing thermal response coefficients. Large values of the thermal response coefficients result in a double-peak nematicon with no oscillations. The linear stability analysis shows that single-peak nematicons and double-peak nematicons having periodic oscillations are stable, while double-peak nematicon with a non-oscillatory nature is unstable.

Onset of absolute instability on a pitching aerofoil

A global transient linear stability analysis of the three-dimensional time-dependent flow around an aerofoil undergoing small-amplitude pitching motion is performed using the optimally time-dependent (OTD) framework. The most salient linear instabilities associated with the instantaneous basic state are computed and tracked over time. The resulting OTD modes reflect the variations in the basic state and can be used as predictors of its spatial and temporal evolution, including the formation of a laminar separation bubble and its gradual spanwise modulation via primary global instability, leading to secondary instability and finally rapid breakdown to turbulence. The study confirms and expands upon earlier stability analyses of the same case based on the local properties of spanwise averaged velocity profiles in the bubble that predicted the onset of absolute instability soon followed by rapid breakdown of the separation bubble. The three-dimensional structure of the most unstable OTD mode is extracted, which compares well with both the locally absolutely unstable mode and the evolution of the basic state itself.