Infinitesimal Perturbation Research Articles

Modulational instability in PT -symmetric Bragg grating structures with saturable nonlinearity

We investigate the nontrivial characteristics of modulational instability (MI) in a system of Bragg gratings with saturable nonlinearity. We also introduce an equal amount of gain and loss into the existing system, which gives rise to an additional degree of freedom, due to the concept of $\mathcal{PT}$ symmetry. We obtain the nonlinear dispersion relation of the saturable model and discover that such dispersion relations for both the conventional and $\mathcal{PT}$-symmetric cases contradict the conventional Kerr and saturable systems by not displaying the typical signature of loop formation in either the upper branch or lower branch of the curve drawn against the wave number and detuning parameter. We employ a standard linear stability analysis to study the MI dynamics of the continuous waves perturbed by an infinitesimal perturbation. The main objective of this paper is twofold: We investigate the dynamics of the MI gain spectrum at the top and bottom of the photonic band gap followed by a comprehensive analysis carried out in the anomalous and normal dispersion regimes. As a result, this perturbed system driven by the saturable nonlinearity and gain or loss yields a variety of instability spectra, which include the conventional sidebands, monotonically increasing gain, the emergence of a single spectrum in either of the Stokes wave-number regions, and so on. In particular, we observe a remarkably peculiar spectrum, which is caused predominantly by the system parameter, though the perturbation wave number boosts the former. We also address the impact of all the physical parameters considered in the proposed model, which include the coupling coefficient, dispersion parameter, and saturable nonlinearity on the phenomenon of MI for different $\mathcal{PT}$-symmetric regimes ranging from the unbroken one to the broken one in greater detail.

Stability of internal gravity wave modes: from triad resonance to broadband instability

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$ , compares to the basic-state amplitude parameter, $\epsilon \ll 1$ . For $\mu ={O}(1)$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

Spectral stability of near-extremal spacetimes

It has been suggested that the spectrum of quasinormal modes of rotating black holes is unstable against additional potential terms in the perturbation equation, as the operator associated with the equation is non-self-adjoint. We point out that a bilinear form has been constructed previously to allow a perturbation analysis of the spectrum, which was applied to study the quasinormal modes of weakly charged Kerr-Newman black holes [Phys. Rev. D 91, 044025 (2015)]. The proposed spectral instability should be restated as instability against potential terms that have an infinitesimal ``energy'' norm that is specifically defined by the type of inner products introduced by Jaramillo et al. and preserving the physical meaning of energy. We argue that it is necessary to address the stability of all previous mode analysis results to reveal their susceptibility to energetically infinitesimal perturbations. In particular, for near-extremal Kerr spacetime, we show that the spectrum of zero-damping modes, which have slow decay rates, is unstable (with order unity fractional change in decay rates) with fine-tuned modification of the potential. The decay rates are, however, always positive with energetically infinitesimal perturbations. If finite potential modifications are allowed near the black hole, it is possible to find superradiantly unstable modes, i.e., a ``black hole bomb'' without an explicit outer shell. For the zero-damping modes in near-extremal Reissner-Nordstr\"om--de Sitter black holes, which are relevant for the breakdown of strong cosmic censorship, we find that the corresponding spectrum is stable under energetically infinitesimal perturbations.

Stability analysis of chaotic systems from data

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.

Onset of oscillatory magnetoconvection under rapid rotation and spatially varying magnetic field

The onset of periodic time-varying magnetoconvection in a regime relevant to the Earth's outer core is investigated in this study. A rapidly rotating plane fluid layer subject to an axially varying horizontal magnetic field is considered under the dynamical regimes of stronger magnetic diffusion compared to thermal and viscous diffusion rates. Dynamically specific convective instabilities, both inside and outside the tangent cylinder regions of the Earth's core, have been investigated by using appropriate patterns of the imposed mean magnetic field. The hallmark of convection onset, with such axially varying mean field, is a viscous oscillatory mode weakly modified by the magnetic field. This modified viscous oscillatory (mVO) mode is observed to exist over a wide range of the strength of the imposed field, making it a dynamically appropriate flow structure with characteristics of outer core convection. An optimal Prandtl number Pr* is found through numerical experiments where the mVO mode can be the most unstable to infinitesimal perturbations. It is further shown that this optimal state admits oscillations for earthlike regimes where thermal diffusion is much less than magnetic diffusion. Also, the formation of columnar convection rolls from isolated vortices is demonstrated as a result of combinations of the classical viscous oscillatory and mVO modes in the rapidly rotating limit (Ekman number E→0). Overall, the qualitative characteristics of magnetoconvection modes for the various imposed patterns are found to be similar despite representing distinct regions in the Earth's outer core.

Bi-stability Bifurcation in Convection Double Diffusion Through a Shallow Horizontal Layer Saturated with a Complex Fluid

The onset of non-linear convection in a porous layer saturated by a shear-thinning liquid is studied. The Carreau-Yasuda model is utilized for modeling the behavior of the working medium. Constant fluxes of heat and mass are defined on the horizontal surfaces of the cavity, while the vertical sides are assumed adiabatic. The parallel flow approximation and the finite difference approach are used to conduct the investigation analytically and numerically, respectively. The linear stability inspection of the convective and diffusive circumstances is carried out by taking into account an infinitesimal perturbation. The theory of linear stability is employed to determine the critical Rayleigh number for the motion from the rest state, Hopf bifurcation, and the transition from the stationary to oscillatory convection. Overall, the Carreau-Yasuda rheological parameters have a significant impact on the thresholds of convection. The most interesting findings of this study is highlighting the existence of a bi-stability phenomenon, i.e., the existence of two steady-state solutions, which was not observed before in non-Newtonian fluids convection.

The near-field shape and stability of a porous plume

When a fluid is injected into a porous medium saturated with an ambient fluid of a greater density, the injected fluid forms a plume that rises upwards due to buoyancy. In the near field of the injection point, the plume adjusts its speed to match the buoyancy velocity of the porous medium, either thinning or thickening to conserve mass. These adjustments are the dominant controls on the near-field plume shape, rather than mixing with the ambient fluid, which occurs over larger vertical distances. In this study, we focus on the plume behaviour in the near field, demonstrating that for moderate injection rates, the plume will reach a steady state, whereby it matches the buoyancy velocity over a few plume width scales from the injection point. However, for very small injection rates, an instability occurs in which the steady plume breaks apart due to the insurmountable density contrast with the surrounding fluid. The steady shape of the plume in the near field depends only on a single dimensionless parameter, which is the ratio between the inlet velocity and the buoyancy velocity. A linear stability analysis is performed, indicating that for small velocity ratios, an infinitesimal perturbation can be constructed that becomes unstable, whilst for moderate velocity ratios, the shape is shown to be stable. Finally, we comment on the application of such flows to the context of CO $_2$ sequestration in porous geological reservoirs.

On the associativity of 1-loop corrections to the celestial operator product in gravity

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. We analyse a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra, however at 1-loop its operator products are corrected by the effective graviton vertex. We argue that the chiral algebra can be interpreted as the universal holomorphic surface defect in the twistor uplift of self-dual gravity, and show that the same correction is induced by an anomalous diagram in the bulk-defect system. The 1-loop holomorphic collinear singularities do not form a consistent chiral algebra. The failure of associativity can be traced to the existence of a recently discovered gravitational anomaly on twistor space. It can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.

On the Discontinuity of Extreme Exponents of Oscillation on a Set of Linear Homogeneous Differential Systems

In this paper, we study the questions of the discontinuity of the extreme oscillation exponents on the set of linear homogeneous differential systems with continuous coefficients on the positive axis. The existence of points on a set of differential systems is established in which all the higher and lower exponents of the oscillation zeros, roots, and hypercorns are not only not continuous, but are not continuous either from above or from below. Moreover, the non-invariance of the extreme exponent of oscillations with respect to infinitesimal perturbations has been proved. When proving the results of this work, the cases of parity and odd order of the matrix of the system are considered separately.

Photo-induced stress relaxation in reconfigurable disulfide-crosslinked supramolecular films visualized by dynamic wrinkling

Stress relaxation in reconfigurable supramolecular polymer networks is strongly related to intermolecular behavior. However, the relationship between molecular motion and macroscopic mechanics is usually vague, and the visualization of internal stress reflecting precise regulation of molecules remains challenging. Here, we present a strategy for visualizing photo-driven stress relaxation induced by infinitesimal perturbations in the intermolecular exchange reaction via reprogrammable wrinkle patterns. The supramolecular films exhibit visible changes in microscopic wrinkle topography through ultraviolet (UV)-induced dynamic disulfide exchange reaction. In accordance with the trans-scale theoretical models, which quantitatively evaluate the chemical-dependent mechanical stresses in the supramolecular network, the unexposed disordered wrinkles evolved into highly oriented patterns and underwent subsequent mutations after thermal treatment. The stress-sensitive wrinkle macro-patterns can be repetitively written/erased through network topology rearrangement using different stimuli. This strategy provides an approach for visualizing and understanding the molecular behavior from dynamic chemistry to mechanical changes, and directly programming wrinkle patterns with regulated structures.

On the Stability of Surface Growth: The Effect of a Compliant Surrounding Medium

AbstractIn a recent paper Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) concerning the stability of surface growth of a pre-stressed elastic half-space with surface tension, it was shown that steady growth is never stable, at least not for all wave numbers of the perturbations, when the growing surface is traction-free. On the other hand, steady growth was found to be always stable when growth occurred on a flat frictionless rigid support and the stretch parallel to the growing surface was compressive. The present study is motivated by these somewhat unexpected and contrasting results.In this paper the stability of a pre-compressed neo-Hookean elastic half-space undergoing surface growth under plane strain conditions is studied. The medium outside the growing body resists growth by applying a pressure on the growing surface. At each increment of growth, the incremental change in pressure is assumed to be proportional to the incremental change in normal displacement of the growing surface. It is shown that surface tension stabilizes a homogeneous growth process against small wavelength perturbations while the compliance of the surrounding medium stabilizes it against large wavelength perturbations. Specifically, there is a critical value of stretch,$\lambda _{\mathrm{cr}} \in (0,1)$λcr∈(0,1), such that growth is linearly stable against infinitesimal perturbations ofarbitrarywavelength provided the stretch parallel to the growing surface exceeds$\lambda _{\mathrm{cr}}$λcr. Thisstability threshold,$\lambda _{\mathrm{cr}}$λcr, is a function of the non-dimensional parameter$\sigma \kappa /G^{2}$σκ/G2, which is the ratio between two length-scales$\sigma /G$σ/Gand$G/\kappa $G/κ, where$G$Gis the shear modulus of the elastic body,$\sigma $σis the surface tension, and$\kappa $κis the stiffness of the surrounding compliant medium.It is shown that$(a)$(a)$\lambda _{\mathrm{cr}} \to 1$λcr→1as$\kappa \to 0$κ→0and$(b)$(b)$\lambda _{\mathrm{cr}} \to 0^{+}$λcr→0+as$\kappa \to \infty $κ→∞, thus recovering the results in Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) pertaining to the respective limiting cases where growth occurs$(a)$(a)on a traction-free surface and$(b)$(b)on a frictionless rigid support. The results are also generalized to include extensional stretches.

Shear-imposed falling thin Newtonian film over a porous slippery surface

The stability of a Newtonian thin film flow over a porous slippery wall approximated by Darcy's law is investigated. The modified Orr–Sommerfeld system is derived for the frequency-dependent linear stability analysis and energy-budget analysis. Moreover, in the longwave regime, both linear and weakly nonlinear stability analyses are conducted for small aspect ratios. In addition, the multiple scale approach is performed directly in the nonlinear deformation equation of the free surface to predict the extraordinary behavior of the amplitude and speed of the nonlinear disturbance in the subcritical and supercritical regimes. The study finds that the larger slip-velocity and externally imposed shear on the thin film increase the total kinetic energy of the infinitesimal perturbations. In a longwave regime, the critical conditions of the primary instability are described as a function of imposed shear stress that destabilizes the film flow for low critical Reynolds number. Furthermore, in the supercritical stable zone, both the nonlinear wave amplitude and phase speed increase with an increase in induced shear in the flow direction and velocity slip, and a reverse trend is observed in applying the imposed shear in the opposite flow direction. On the other hand, the nonlinear wave amplitude in the subcritical unstable zone increases and decreases, corresponding to the larger values of imposed shear and slip parameters, respectively.

Copula sensitivity analysis for portfolio credit derivatives

Modeling dependence among input random variables is often critical for performance evaluation of stochastic systems, and copulas provide one approach to model such dependence. In the financial industry, the dependence among default times of different assets drives the pricing of portfolio credit derivatives, including basket default swaps and collateralized debt obligations. Copula sensitivities provide information about how changes in the dependence level affect the output. We study sensitivity analysis of elliptical copulas and Archimedean copulas using infinitesimal perturbation analysis, and an unbiased estimators is derived using conditional Monte Carlo (CMC) to address discontinuities that arise in portfolio credit derivatives. In simulation experiments, the new estimators have smaller variance than other applicable methods.

Reduced modelling and global instability of finite-Reynolds-number flow in compliant rectangular channels

Experiments have shown that flow in compliant microchannels can become unstable at a much lower Reynolds number than the corresponding flow in a rigid conduit. Therefore, it has been suggested that the wall's elastic compliance can be exploited towards new modalities of microscale mixing. While previous studies mainly focused on the local instability induced by the fluid–structure interactions (FSIs) in the system, we derive a one-dimensional (1-D) model to study the FSI's effect on the global instability. The proposed 1-D FSI model is tailored to long, shallow rectangular microchannels with a deformable top wall, similar to the experiments. Going beyond the usual lubrication flows analysed in these geometries, we include finite fluid inertia and couple the reduced flow equations to a novel reduced 1-D wall deformation equation. Although a quantitative comparison with previous experiments is difficult, the behaviours of the proposed model show, qualitatively, agreement with the experimental observations, and capture several key effects. Specifically, we find the critical conditions under which the inflated base state of the 1-D FSI model is linearly unstable to infinitesimal perturbations. The critical Reynolds numbers predicted are in agreement with experimental observations. The unstable modes are highly oscillatory, with frequencies close to the natural frequency of the wall, suggesting that the observed instabilities are resonance phenomena. Furthermore, during the start-up from an undeformed initial state, self-sustained oscillations can be triggered by FSI. Our modelling framework can be applied to other microfluidic systems with similar geometric scale separation under different operating conditions.

Sensitivity of low-rank matrix recovery

We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. This is one customary approach to build recommender systems. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of measurements affects it. In addition, we study the condition number of the best rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for its condition number, which shows that it depends on the relative singular value gap between the rth and $$(r+1)$$ th singular values of the input matrix.

Quantum sensing with sub-Planck structures for the dynamics of Bose-Einstein condensate in presence of engineered potential barriers inside a harmonic trap

We report a scheme for quantum sensing in ultracold atoms by utilizing an atom-interferometer producing sub-Planck scale structures. The condensate is trapped in a hybrid potential, comprising of an overall harmonic trap and sharp potential barriers which are designed as 50-50 atomic beam-splitters. A scaling law for quantum sensing is established for ultracold atoms and the maximum limit of quantum sensing is identified. Quantum sensing for position, momentum and temperature are demonstrated with a sufficiently stable condensate. The present scheme reveals maximum limit of quantum sensitivity for ultracold atoms against infinitesimal external perturbations. With a designable time sequence for the trapping potential, our results provide a promising scheme for quantum sensing with Bose-Einstein condensate in atomic chips.

Phase transitions as a manifestation of spontaneous unitarity violation

Spontaneous symmetry breaking is well understood under equilibrium conditions as a consequence of the singularity of the thermodynamic limit. How a single global orientation of the order parameter dynamically emerges from an initially symmetric state during a phase transition, however, is not captured by this paradigm. Here, we present a series of symmetry arguments suggesting that singling out a global choice for the ordered state is in fact forbidden under unitary time evolution, even in the presence of an environment and infinitesimal symmetry breaking perturbations. We thus argue that the observation of phase transitions in our everyday world presents a manifestation of the unitarity of quantum dynamics itself being spontaneously broken. We argue that this agrees with the observation that Schrödinger’s time dependent equation is rendered unstable for macroscopic objects owing to the same singular thermodynamic limit that affects equilibrium configurations.

Stability of Multidegree of Freedom Potential Systems to General Infinitesimal Positional Perturbation Forces

Abstract The stability of general linear multidegree of freedom stable potential systems that are perturbed by general arbitrary positional forces, which may be neither conservative nor purely circulatory/conservative, is considered. It has been recently recognized that such perturbed potential systems with multiple frequencies of vibration are susceptible to instability, and this paper is centrally concerned with the situation when potential systems have such multiple natural frequencies. An approach based on perturbation theory that includes nonlinear terms in the expansions of the perturbed eigenvalues is developed. Explicit conditions under which the system either remains stable or becomes unstable due to flutter are provided. These results show that the stability/instability picture that emerges is far subtler and more complex than what might be intuitively inferred. The manner in which prior results related to narrower classes of perturbation matrices, like circulatory matrices, get included in the more general results obtained here is pointed out. Several numerical examples illustrate the applicability of the analytical results. An engineering application is provided demonstrating the power of the analytical results.

Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall

Plane Poiseuille flow of a Bingham fluid in a channel armed with a superhydrophobic (SH) lower wall is analysed via a semi-analytical model, accompanied by complementary direct numerical simulations (DNS). The SH surface represents a groovy structure with air trapped inside its cavities. Therefore, the fluid adjacent to the wall undergoes stick–slip conditions. The model is developed based on introducing infinitesimal wall-induced perturbations into the motion equations, followed by Fourier series expansions, and solving the resulting equations as a boundary value problem. The Navier slip law accounts for the slip at the liquid/air interface (assuming the Cassie state). The presented analysis is fairly comprehensive, covering the creeping and inertial regimes for thick channels (via the semi-analytical and DNS solutions). The main dimensionless numbers are the Reynolds ( $Re$ ), Bingham ( $Bi$ ) and slip ( $b$ ) numbers, as well as the groove periodicity length ( $\ell$ ) and the slip area fraction ( $\varphi$ ). By increasing $Bi$ , the perturbation and slip velocity fields grow. As $Re$ increases, the perturbation and slip velocity fields become asymmetric. For certain flow parameters, an unyielded plug zone may appear on the SH wall liquid/air interface, while its formation is accelerated by inertial effects. The results classify the regimes of creeping and inertial flows via predicting the onset of the unyielded plug zone formation at the SH wall.

Optimal Linear Response for Markov Hilbert–Schmidt Integral Operators and Stochastic Dynamical Systems

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an L^2 kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau–Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.