Wave Hierarchy Research Articles

A simple predictive three-step chemical model for gaseous stoichiometric hydrogen–oxygen detonation quenching

Two types of three-step chain-branching kinetics for gaseous hydrogen–oxygen detonation quenching are proposed. The reaction rates dependency on density and pressure, as well as the change in molecular weight are included and fitted to results of detailed chemistry not only along the Hugoniot curve but also behind transverse waves. The trend of induction and reaction times and reduced activation energy of zero-dimensional constant pressure combustion process as well as the induction, reaction lengths, and peak thermicity of the ZND model are focused on in the calibration procedure. One model (3SM-I) is designed to model the ideal detonation whereas the other (3SM-N) focuses on the Dn−κ curve during calibration, i.e. the detonation velocity-mean front curvature curves, aiming to capture the non-idealities. The predictive ability using the proposed models is evaluated in a detonation transmission configuration, a layer of ▪ mixture being confined by N2, in determining the critical height, reactive layer height at which detonation transmission is not possible. Both approaches get closer to the critical height from detailed chemistry. With the present set of model parameters, 3SM-I is better than 3SM-N for prediction of critical height. Moreover, borrowing analytical ignition criteria and wave hierarchy approach, the comparison of various chemical models that are able to recover the critical height suggests that the slope of the Dn−κ curve, linked with the chemical response to losses, is the first key parameter for the critical height determination. In addition, the dynamics of the extinction near criticality such as appearance of transverse detonation and quenching distance from 3SM-I are in very good agreement with that of detailed chemistry due to better estimation of the chemical features along the Hugoniot curve and behind transverse waves. The present study provides a guideline to get simplified chemical models from detailed chemistry for predictive simulations.Novelty and significance statementThe novelty of research is two-fold. The first is to provide general guideline to get simplified models from detailed chemistry for the simulation of detonation extinction with improved predictive ability. The tested configuration was that of a stoichiometric hydrogen–oxygen mixture semi-confined by nitrogen. The evolution of the induction, reaction times, and the maximum thermicity were used as fitting targets for our 3-step model. Furthermore, they were evaluated not only at the post-shock states along the Hugoniot curve, but also behind the transverse waves, keystones of the cellular structure. Second, borrowing analytical ignition criteria and wave hierarchy approach, the comparison of different chemical models that were able to recover the critical height suggested that the slope of the normal velocity–curvature curve was the first key parameter for the critical height determination. Moreover, the dynamics of extinction near criticality, and quenching distance depended on the reaction rate dependency on density and pressure.

On the N-waves hierarchy with constant boundary conditions. Spectral properties

This paper is devoted to [Formula: see text]-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators [Formula: see text], whose potentials [Formula: see text] tend to constants [Formula: see text] for [Formula: see text]. For special choices of [Formula: see text], we outline the spectral properties of [Formula: see text], the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the [Formula: see text]-derivative of the potential [Formula: see text]. Next, using the Wronskian relations, we derive the dispersion laws for the [Formula: see text]-wave hierarchy and describe the NLEE related to the given Lax operator.

On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium

In the proposed paper, the process of propagation of shock waves in two-dimensional media without its own pressure drop is studied. The model of such media is a system of equations of gas dynamics, where formally the pressure is assumed to be zero. From the point of view of the theory of systems of conservation laws, the system of equations under consideration is in some sense degenerate, and, consequently, the corresponding generalized solutions have strong singularities (evolving shock waves with density in the form of delta functions on manifolds of different dimensions). We will denote this property as the evolution of the hierarchy of strong singularities or the evolution of the hierarchy of shock waves. In the paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta function along curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy of shock waves arises. The properties of such dynamics of strong singularities are described. The results obtained provide a starting point for moving on to a much more interesting multidimensional case in the future.

On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium

Изучается процесс распространения ударных волн в двумерных средах без собственного перепада давления. Модель таких сред представляет собой систему уравнений газовой динамики, где формально давление положено равным нулю. С точки зрения теории систем законов сохранения рассматриваемая система уравнений является в некотором смысле вырожденной, и, вследствие этого, соответствующие обобщенные решения обладают сильными особенностями: эволюционирующими ударными волнами с плотностью в виде дельта-функций на многообразиях разной размерности. Это свойство будем обозначать как эволюцию иерархии сильных особенностей или эволюцию иерархии ударных волн. В двумерном случае доказано существование такого взаимодействия сильных особенностей с дельта-функцией плотности вдоль кривых в пространстве $\mathbb{R}^2$, при котором возникает концентрация плотности в точке, т. е. возникает иерархия ударных волн. Описаны свойства подобной динамики сильных особенностей. Полученные результаты являются отправной точкой для перехода в дальнейшем к гораздо более интересному многомерному случаю. Библиография: 44 наименования.

On the Quartic Korteweg–de Vries hierarchy of nonlinear Rossby waves and its dynamics

A long-standing challenge in exploring the dynamics of large-scale atmospheric or oceanic motions is to find more appropriate theoretical models. This paper aims at the mechanisms of nonlinear Rossby waves by a new obtained nonlinear evolutionary equation hierarchy approach. The multiple scales method and weak nonlinear perturbation method are adopted based on the quasi-geostrophic potential vorticity conservation law for large scale atmospheric or oceanic motions. It is the first time that a Quartic Korteweg–de Vries model equation is derived to describe the propagation of nonlinear Rossby waves amplitude, which provides a new acceptable theoretical model for the study of atmospheric blocking phenomena. Qualitative and quantitative analysis for the new Quartic KdV model are finished to disclose the evolutionary mechanisms of nonlinear Rossby waves. The corresponding results declares that the β effect, topographic features and detuning parameter are all important factors on the nonlinear Rossby waves.

The hierarchy of coupled sleep oscillations reverses with aging in humans.

A well-orchestrated coupling hierarchy of slow waves and spindles during slow wave sleep supports memory consolidation. In old age, duration of slow wave sleep and number of coupling events decreases. The coupling hierarchy deteriorates, predicting memory loss and brain atrophy. Here, we investigate the dynamics of this physiological change in slow wave-spindle coupling in a frontocentral electroencephalography position in a large sample (N=340, 237 female, 103 male) spanning most of the human lifespan (ages 15-83). We find that, instead of changing abruptly, spindles gradually shift from being driven by-, to driving slow waves with age, reversing the coupling hierarchy typically seen in younger brains. Reversal was stronger the lower the slow wave frequency, and starts around midlife (∼age 40-48), with an established reversed hierarchy at age 56-83. Notably, coupling strength remains unaffected by age. In older adults, deteriorating slow wave-spindle coupling, measured using phase slope index (PSI) and number of coupling events, is associated with blood plasma glial fibrillary acidic protein (GFAP) levels, a marker for astrocyte activation. Data-driven models suggest decreased sleep time and higher age lead to fewer coupling events, paralleled by increased astrocyte activation. Counterintuitively, astrocyte activation is associated with a back-shift of the coupling hierarchy (PSI) towards a "younger" status along with increased coupling occurrence and strength, potentially suggesting compensatory processes. As the changes in coupling hierarchy occur gradually starting at midlife, we suggest there exists a sizable window of opportunity for early interventions to counteract undesirable trajectories associated with neurodegeneration.Significance StatementEvidence accumulates that sleep disturbances and cognitive decline are bi-directionally and causally linked forming a vicious cycle. Improving sleep quality could break this cycle. One marker for sleep quality is a clear hierarchical structure of sleep oscillations. Previous studies showed that sleep oscillations decouple in old age. Here, we show that, rather, the hierarchical structure gradually shifts across the human lifespan and reverses in old age, while coupling strength remains unchanged. This shift is associated with markers for astrocyte activation in old age. The shifting hierarchy resembles brain maturation, plateau, and wear processes. This study furthers our comprehension of this important neurophysiological process and its dynamic evolution across the human lifespan.

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Nonlinear dynamics of a liquid film flow over a solid substrate in the presence of external shear stress and electric field

Using approximation techniques, long-wave length framework and boundary-layer, the effects of electrostatic force and induced shear stress on the flow behavior down an inclined solid substrate are investigated. In general case, the considered model accounts in the presence of inertia regime and streamwise viscous diffusion with the influence of normal electric field and an imposed shear stress. Using the Galerkin weighted residual, two coupled evolution equations for the flow rate and film thickness are extracted. In the appropriate limit cases, the evolution equations obtained by previous authors are recovered. The primary instability has been analyzed using the Whitham wave hierarchy framework. In the nonlinear regime, the behavior of solitary waves arose on the surface of liquid film due to the effects of electrostatic force and imposed shear stress throughout three-dimensional dynamical systems. Some bifurcation points are reported. In both extremely viscous and electrogravity regimes, the Benney-like equation is extracted in a new form. By excluding contribution of external shear stress and viscous dispersion parameter, the interesting results of previous authors are recovered. In both weakly nonlinear and inertialess regimes, the bifurcation points of the three dynamical systems are discussed within the Kuramoto–Sivashinsky type equation.

Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals

We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.

Stability of a thin viscoelastic film falling down an inclined plane

We study the long-wave instability of a falling film of the Oldroyd-B fluid. The weighted residual method is adopted to model the film dynamics in both linear and nonlinear regimes. The mechanism of the viscoelastic effect on the instability is explored within the framework of the Whitham wave hierarchy theory. We further analyze the film evolution and the traveling waves by means of nonlinear simulations.

Algebro-geometric integration of a modified shallow wave hierarchy

Abstract By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

Liouville reflection operator, affine Yangian and Bethe ansatz

In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik Okounkov R-matrix. We discuss relation between RLL and current realization of the affine Yangian of mathfrak{gl}(1) . We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.

Analytical solution to one-dimensional mathematical model of flow and morphological evolution in open channels

The evolution of open-channel flow and morphology can be simulated by one-dimensional (ID) mathematical models. These models are typically solved by numerical or analytical methods. Because the behavior of variables can be explained by explicit mathematical determinations, compared to numerical solutions, analytical solutions provide fundamental and physical insights into flow and sediment transport mechanisms. The singular perturbation technique derives a hierarchical equation of waves and describes the evolutionary nature of disturbances in hyperbolic systems. The wave hierarchy consists of dynamic, diffusion, and kinetic waves. These three types of waves interact with each other in the process of propagation. Moreover, the Laplace transform is implemented to transform partial differential equations into ordinary differential equations. Analytical expressions in the wave front are subtracted by the approximation of kinetic and diffusion models. Moreover, an analytical solution consists of a linear combination of the kinetic wave front and the diffusion wave front expressions, pursuing to describe the physical mechanism of motion in open channels as completely as possible. Analytical solutions are presented as a combination of exponential functions, hyperbolic functions, and infinite series. The obtained analytical solution was further applied to the simulation of flood path and morphological evolution in the Lower Yellow River. The phenomenon of increased peak discharge in the downstream reach was successfully simulated. It was encouraging that the results were in good agreement with the observed data.

On matrix fourth Painlevé hierarchies

We define new matrix fourth Painlevé hierarchies and a new matrix second Painlevé hierarchy. For our matrix fourth Painlevé hierarchies, properties such as auto-Bäcklund transformations, Bäcklund transformations and special integrals are derived, and known relations between fourth Painlevé transformations are extended to the matrix ODE hierarchies. The matrix ODE hierarchies presented here are derived as reductions of a new nonisospectral matrix dispersive water wave hierarchy and modifications thereof.

Liouville correspondences between multicomponent integrable hierarchies

In this paper, we establish Liouville correspondences for the integrable two-component Camassa-Holm hierarchy, the two-component Novikov (Geng-Xue) hierarchy, and the two-component dual dispersive water wave hierarchy by means of the related Liouville transformations. This extends previous results on the scalar Camassa-Holm and KdV hierarchies, and the Novikov and Sawada-Kotera hierarchies to the multi-component case.

Thermal instability revisited

ABSTRACT Field’s linear analysis of thermal instability is repeated using methods related to Whitham’s theory of wave hierarchies, which brings out the physically relevant parameters in a much clearer way than in the original analysis. It is also used for the stability of non-equilibrium states and we show that for gas cooling behind a shock, the usual analysis is only quantitatively valid for shocks that are just able to trigger a transition to the cold phase. A magnetic field can readily be included and we show that this does not change the stability criteria. By considering steady shock solutions, we show that almost all plausible initial conditions lead to a magnetically dominated state on the unstable part of the equilibrium curve. These results are used to analyse numerical calculations of perturbed steady shock solutions and of shocks interacting with a warm cloud.

Балтійсько-Іранський суперлінеамент — довгоживучий транс євразійський пояс дислокацій і планетарної мегатріщинуватості

Information on fault tectonics and the deep structure of the global trans-Eurasian belt of dislocations and planetary megafractures (Baltic-Iranian Super Lineament) is generalized based on the analysis of literary, geological and geophysical, cartographic, cosmographic materials, including the results of tectonic lineaments interpretation of cosmo-images of the Earth’s surface. The features of its distribution and segmentation are clarified. This Super Lineament is the trans-Eurasian structure of a planetary rank — an integral part of the regularly spatially oriented ancient reghmatic fault net of Eurasia, formed at the early stages of the formation of the rigid earth crust under the influence of global stresses associated with the factor of rotation of the Earth. The long-lived nature of the belt (the Upper Proterozoic — Cenozoic) and its quasi-stationary position throughout its formation history, as well as the absence of large-scale, hundreds of kilometers, of shear displacements along it and at intersections with transverse tectolineament belts of a similar rank indicate the low probability of the plate-tectonic convergence model of Afro-Arabia and Eurasia. The destruction model of the once united Afro-Eurasian continent in a potentially shear field of relatively low amplitude (on a planetary scale) dislocations is preferable. The geodynamic mode of formation and development of the belt as a whole cannot be described by any one type — riftogenic, or collisional ones. Regional and supra-regional zones of extension (riftogenic zones) and compression (collision) alternate in it periodically. It shows the global nature of the belt, the union of different geodynamic regimes in it, as well as the wave character and hierarchy of the tectonic process.

Some symmetries and similarity solutions of the long-water wave hierarchy

We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

Rogue Wave and Breather Structures with “High Frequency” and “Low Frequency” in 𝒫𝒯-Symmetric Nonlinear Couplers with Gain and Loss

Abstract Based on the modified Darboux transformation method, starting from zero solution and the plane wave solution, the hierarchies of rational solutions and breather solutions with “high frequency” and “low frequency” of the coupled nonlinear Schrödinger equation in parity-time symmetric nonlinear couplers with gain and loss are constructed, respectively. From these results, some basic characteristics of multi-rogue waves and multi-breathers are studied. Based on the property of rogue wave as the “quantum” of pattern structure in rogue wave hierarchy, we further study the novel structures of the superposed Akhmediev breathers, Kuznetsov-Ma solitons and their combined structures. It is expected that these results may give new insight into the context of the optical communications and Bose-Einstein condensations.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations**Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.