Instability Of Traveling Waves Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Coherent flow structures and magnetic field patterns in rotating spherical shell convective dynamos: A data-driven approach

The Earth's outer core dynamics involve convective fluid motion generating an observable geomagnetic field. The velocity and magnetic fields exhibit characteristic spatiotemporal features possessing geophysical significance for which extensive datasets are available from direct observations and computational simulations. This study demonstrates the robustness of proper orthogonal decomposition (POD), a data-driven technique, in detecting prominent and relevant features in these datasets. Improvising on previous practices, the POD efficiently detects infinitesimal instabilities at the onset of convection, providing an accurate and objective methodology to determine the convective threshold, even for heterogeneous buoyancy forcing. Time evolution of paired, phase-shifted modes efficiently reconstructs the azimuthally drifting of traveling wave instabilities. Simultaneously reduced order modeling of velocity components clearly distinguish the equatorial and polar coherent flow structures. Supercritical convection-driven magnetic field data over long periods, generated using numerical simulations, produce dominant modes that are more accurately representative of time-averaged patterns than geocentric axial dipole patterns. Moreover, the quantitative significance of the dominant modes determines the extent of dimensional reduction complementing established diagnostics for dipolarity. Finally, analysis of observational geomagnetic field data reveals long-lived dominant patterns influenced by thermal core–mantle interaction consistent with numerical models employing tomographic heat flux boundary conditions in present as well as previous studies.

Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation

Transversal spectral instability of periodic traveling waves for the generalized Zakharov–Kuznetsov equation

Isolas of localized structures and Raman–Kerr frequency combs in micro-structured resonators

We theoretically investigate the combined impact of the Kerr and stimulated Raman scattering effect on the formation of localized structures and frequency comb generation. We focus on the regime of traveling wave instability. We first perform a real-order parameter description by deriving a Swift–Hohenberg equation with nonlocal delayed feedback. Second, we characterize the motion of traveling wave periodic solutions by estimating their thresholds as well their speed. By using a numerical continuation method, we construct a bifurcation diagram showing the emergence of traveling wave periodic solutions, as well as bright and dark moving localized structures. Numerical simulations of the generalized Lugiato–Lefever equation confirm evidence of isolas of localized structures. More importantly, we show that the stimulated Raman scattering strongly impacts the dynamics of localized structures by creating isolas consisting of bright and dark localized structures, and by inducing a motion of these structures. Finally, we provide a geometrical interpretation of the formation of isola stacks based on dynamical systems theory.

On convective instabilities in a rotating fluid with stably stratified layer and thermally heterogeneous boundary

The onset of convection in a rotating plane layer due to a vertical temperature gradient is studied in this paper. The background stratification is modulated by lateral temperature variations and stable stratification aimed at understating the Earth's outer core convection subject to thermal core–mantle interaction. At the top boundary, sinusoidal and Gaussian temperature variations are imposed apart from the reference case of isothermal condition used in the classical Rayleigh–Benard convection. The additional modulating conditions break the top–bottom flow symmetry leading to flow localization and asymmetry that exhibit modified temporal dynamics unlike that of the classical Rayleigh–Benard cells. The threshold for convection is lowered with flows occurring in surplus heat flux regions caused by the imposed conditions. Despite flow suppression in the stable layer, rapid rotation favors the penetration of convection rolls with smaller wavelengths. The lateral variations in temperature imposed at the top boundary enhance such axial penetration with a laterally varying penetrative extent resulting in a modified clustered flow structure unlike the reference case. With both modulating conditions imposed, the onset of overstable modes is favored for low Prandtl numbers, a regime which is relevant to the Earth's core conditions. With rapid rotation, a novel mode of traveling wave instability occurs at the onset of convection, the propagation direction of which is controlled by the lateral temperature gradients at the top boundary. The onset of oscillatory modes is suppressed by the imposition of the modulating conditions indicated by the significant lowering of the transition Prandtl number.

Superharmonic instability for regularized long-wave models* *JCB is supported by NSF DMS-1615418. VMH is supported by NSF DMS-2009981.

We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney–Luke, and Benjamin–Bona–Mahony equations. Of particular interest is a striking new instability phenomenon—spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg–de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We perform rigorous spectral asymptotics for short wavelength perturbations to establish conditions under which the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments which corroborate our analytical findings.

Transverse spectral instability in generalized Kadomtsev–Petviashvili equation

We study transverse stability and instability of one-dimensional small-amplitude periodic travelling waves of a generalized Kadomtsev–Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obtain transverse stability results in the aforementioned equations.

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

Determination of groove shape with strong destabilization and low hydraulic drag

Flow through a channel equipped with plane, longitudinal grooves is investigated. We focus on determining changes to the flow dynamics due to applied wall manipulations, especially the possibility of drag-reduction, potential for hydrodynamic destabilization and onset of secondary, nonlinear flow solutions. Considered patterns of geometrical manipulation consist of plane walled grooves of triangular, variations of trapezoidal and up to rectangular shapes and are compared with existing results obtained for channels with sinusoidal corrugations. We show that there exists a strong connection in the response of the flow system to the applied pattern of grooves with the leading Fourier component of the wall pattern. The analysis starts with undisturbed, two-dimensional flows investigated with the focus on hydraulic resistance for wide range of geometric parameters. Secondly, critical conditions for the onset of travelling wave instability are determined and compared with existing results. Thirdly, nonlinear flow solutions, obtained for flows through the geometry with lowest destabilization threshold are analyzed at supercritical conditions. Finally quantification of the diffusive transport intensification due to the kinematics of the nonlinear flow solution is attempted. It is shown that, irrespective of the groove shape, longitudinal wall patterns result in flow destabilization due to traveling wave mode already at very low values of the Reynolds number (<102). At the same time such configurations can be energy efficient since overall drag is marginally increased (or in fact reduced) and at the same time diffusive processes are intensified leading to improved mixing. Implications of this study might help in development of small-scale flow devices operating at low and moderate ranges of Reynolds number with the purpose of intensifying mixing, heat transfer or reaction of chemical or biological compounds.

Suppression of wave instability in broad-area lasers by the external optical injection

We theoretically investigate suppression of travelling wave instability in the transverse section of broad-area lasers. We investigate influence of the detuning between injected signal and generated radiation. The synchronization conditions were obtained under which the effective suppression of the wave instability occurs. Numerical simulation of the spatiotemporal dynamics of a broad-area laser with injection of external optical signal is carried out.

Stability of periodic traveling flexural‐gravity waves in two dimensions

AbstractIn this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.

Flow dynamics in longitudinally grooved duct

Flow in a finite-width rectangular duct with a corrugated top-bottom wall has been studied. The primary goal is to establish geometries that allow early flow destabilization at a possibly low drag increase. The flow is assumed periodic in the streamwise direction and bounded by the duct sidewalls in the spanwise direction; the top and bottom wall corrugations have a form of sinusoidal waves oriented transversely to the flow and form longitudinal grooves; i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional base flow. Mainly, the dependence of hydraulic losses and drag reducing properties on duct’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional travelling wave instability over a wide spectrum of geometric configurations. Linear stability is investigated by means of the direct numerical simulation of the Navier-Stokes equations. Critical conditions for the onset of instabilities at a range of geometric parameters are determined. Finally, the nonlinear saturation of unstable modes and the resulting secondary flows are examined. We have shown that in the state resulting from the nonlinear saturation of the disturbance, the flow becomes more complex while the drag reducing properties of the base flow can be maintained.

A diffusion-flame analog of forward smolder waves: (II) stability analysis

We proceed to examine the stability of the adiabatic and non-adiabatic structures of forward smolder waves elaborated in Part (I) of this series. The dispersion relation for adiabatic forward smolder waves with a reaction trailing structure turns out to take a form similar to that for premixed flames, thereby strengthening the analogy of the reaction trailing structure with the premixed flame regime of diffusion flames. According to the dispersion relation, corresponding to each Damköhler number there exists a marginal oxygen Lewis number, below which cellular instability occurs. In particular, similar to the Burke–Schumann limit of diffusion flames, the stoichiometric limit at infinite Damköhler number is unconditionally stable. Such unconditional stability is found to further extend to the entire Damköhler number range for adiabatic forward smolder waves with a reaction leading structure. Linear stability analysis of non-adiabatic forward smolder waves indicates that, for both reaction trailing and reaction leading structures, the low smolder temperature (or high reactant leakage) solution branch is physically unrealistic, whereas on the high smolder temperature (or low reactant leakage) branch different kinds of instabilities may develop near the quenching limit. Under a fixed Damköhler number, the range of the heat loss coefficient corresponding to these instabilities shows a trend to grow with decreasing oxygen Lewis number. 2-D time-dependent numerical simulations of unstable non-adiabatic forward smolder waves confirm that fingering or cellular instability occurs exclusively for the reaction trailing structure, whereas traveling wave instability prevails for the reaction leading structure. A comparison is made between the current stability analysis results of non-adiabatic forward smolder waves and results from a concurrent flame spread experiment. Agreement is achieved not only on the existence of reaction front instabilities near the quenching limit, but also on the conditions determining the type of these instabilities.

Direct numerical simulations of creeping to early turbulent flow in unbaffled and baffled stirred tanks

It has been known for a long time that the fluid flow and several global quantities, such as the power and pumping numbers, are about the same in baffled and unbaffled mechanically stirred vessels at low Reynolds numbers, but bifurcate at some intermediate Re and take drastically different values in fully turbulent flow. However, several details are not yet completely understood, notably concerning the relation of this bifurcation with the flow features and the transition to turbulence. In order to shed light on these issues, computational fluid dynamics was employed to predict the flow field in two vessels stirred by a six-bladed Rushton turbine at Reynolds numbers from 0.2 to 600 (covering the range from creeping flow to early turbulent flow). The two vessels differed only for the presence or absence of peripheral baffles. All simulations were conducted by a finite volume method in time-dependent mode, and a sliding-mesh technique was used in the baffled case to deal with the relative motion of baffles and impeller blades. A sensitivity analysis proved that a grid of about 5 million finite volumes was adequate to yield grid-independent results. The study proved that the bifurcation between quantities related to baffled and unbaffled tanks occurs when the inner (near-impeller) and outer (near-wall/baffles) flow fields interact significantly. It also elucidated the mechanisms of transition to turbulence in baffled and unbaffled tanks, notably showing in this latter case the existence (in the rotating reference frame of the impeller) of a periodic flow regime which involves a travelling wave instability.

Spiral breakup in a RD system of cardiac excitation due to front–back interaction

We consider a two-variable partial differential equations model of cardiac excitation and study spiral wave instability in a one-parameter family of solutions. We investigate numerically the existence of periodic traveling wave solution and show the front and the back interaction far away from the bifurcation point in one dimension. In two dimensions, we show the emergence of a stable spiral pattern before the bifurcation point. The most complex spatiotemporal pattern is called ventricular fibrillation when the breakup of one spiral wave makes another wave and the medium becomes chaotic. We show spiral wave instability and periodic traveling wave instability in the same computational settings. It is found that the pattern of the front–back interaction in two dimensions is similar with that of in the one dimension.

Flow transition and hydrothermal wave instability of thermocapillary-driven flow in a free rectangular liquid film

Thermocapillary-driven flow in a free liquid film, which has two gas-liquid interfaces, is experimentally investigated. Silicone oil of 5 cSt is employed as the test fluid. Two-dimensional basic flow known as ‘double-layered flow’ after Ueno and Torii (2010) is realized under small-enough Marangoni number Ma, the non-dimensional number to describe the intensity of thermocapillary effect, under the geometry considered in the present study. The flow exhibits a transition from the two-dimensional steady flow state to the three-dimensional oscillatory state when the intensity of the imposed thermocapillary effect along the free surfaces exceeds the threshold. In this oscillatory regime, two types of hydrothermal wave instabilities are observed: the traveling-wave flow and the standing-wave flow. We especially focus on the traveling-wave instability and compare it with a hydrothermal wave in a liquid layer with a single free surface investigated by other researchers.

Instability in a channel with grooves parallel to the flow

Flow in a channel with corrugated walls has been studied, with the primary goal of establishing channel geometries that enhance achievable mixing at possibly low drag increase. The wall corrugation has the form of a sinusoidal wave oriented transversely, i.e., the lines of constant elevation (or phase) are parallel to the direction of the flow. The analysis is performed up to the Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is focused on the properties of the two-dimensional, base flow. Mainly, the dependence of the drag on the channel’s geometry is characterized. The second part of the analysis discusses the onset of the three-dimensional traveling wave instability. Linear stability is investigated by the Direct Numerical Simulation of the Navier-Stokes equations. Critical conditions for the onset of instabilities at a range of geometric parameters are determined. Finally, nonlinear saturation of the unstable modes and the resulting secondary flows is examined. It is shown that the drag reduction property of the base flow can be maintained in the state resulting from non-linear saturation of the disturbance.

Flow dynamics and enhanced mixing in a converging–diverging channel

An analysis of flows in converging–diverging channels has been carried out with the primary goal of identifying geometries which result in increased mixing. The model geometry consists of a channel whose walls are fitted with spanwise grooves of moderate amplitudes (up to 10 % of the mean channel opening) and of sinusoidal shape. The groove systems on each wall are shifted by half of a wavelength with respect to each other, resulting in the formation of a converging–diverging conduit. The analysis is carried out up to Reynolds numbers resulting in the formation of secondary states. The first part of the analysis is based on a two-dimensional model and demonstrates that increasing the corrugation wavelength results in the appearance of an unsteady separation whose onset correlates with the onset of the travelling wave instability. The second part of the analysis is based on a three-dimensional model and demonstrates that the flow dynamics is dominated by the centrifugal instability over a large range of geometric parameters, resulting in the formation of streamwise vortices. It is shown that the onset of the vortices may lead to the elimination of the unsteady separation. The critical Reynolds number for the vortex onset initially decreases as the corrugation amplitude increases but an excessive increase leads to the stream lift up, reduction of the centrifugal forces and flow stabilization. The flow dynamics under such conditions is again dominated by the travelling wave instability. Conditions leading to the formation of streamwise vortices without interference from the travelling wave instability have been identified. The structure and the mixing properties of the saturated states are discussed.

Impact of time-delayed feedback on spatiotemporal dynamics in the Lugiato-Lefever model

We analyze the impact of delayed optical feedback (OF) on the spatiotemporal dynamics of the Lugiato-Lefever model. First, we carry out linear stability analysis and reveal the role of the OF strength and phase on the shape of the bistable curve as well as on Turing, Andronov-Hopf, and traveling-wave instability regions. Further, we demonstrate how the OF impacts the spatial dynamics by shifting the regions with different spatial eigenvalue spectra. In addition, we reveal a clustering behavior of cavity solitons as a function of the OF strength at fixed OF phase. Depending on the feedback parameters, OF can also induce a drift bifurcation of a stationary cavity soliton, as well as an Andronov-Hopf bifurcation of a drifting soliton. We present an analytical expression for the threshold of the drift bifurcation and show that above a certain value of the OF strength the system enters a region of spatiotemporal chaos.

Intrinsic performance-limiting instabilities in two-level class-B broad-area lasers

The present paper is concerned with the analytical and numerical investigation of the transverse spatio-temporal instabilities in two-level broad-area lasers for the specific class-B case. We show that the two-level class-B broad-area laser tends to naturally operate in the filamentary state. This is revealed to be provided with two causes. First of them is related with the homogeneous output profile being intrinsically unstable due to the traveling-wave instability, independently from the boundary conditions. Secondly, high sensitivity to the boundaries of the pumping region was found for the commonly used top-hat-like profile leading to boundaries-induced filamentary dynamics. Spatio-temporal properties of both instability mechanisms are studied and their effects on the resulting laser dynamical behavior are analyzed.