Dispersion Equation Research Articles

On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

Abstract In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given.

A Study on the Coexistence of Anthropogenic and Natural Sources in a Three-Dimensional Aquifer

A study using mathematical modeling has been conducted to analyze how both man-made and natural sources of contaminants affect various layers of an aquifer-aquitard system. The xy-, yz-, and zx-plane have been used to depict the locations where the natural sources of contaminant occur on the xz- and yz-plane, and where the man-made sources occur, on the xy-plane. It is assumed that the sources occurring in different planes are constant, while the velocity of groundwater flow has been considered only along the x-axis. A three-dimensional advection dispersion equation (ADE) has been used to accurately model the flow of groundwater and contaminants through a porous medium. Three distinct sources exert their influence on three separate planes throughout the entire duration of this study, thus making it possible to model these sources using initial conditions. This study presents a profile of contaminant concentration in space and time when constant sources are located on different planes. Some physical assumptions have been considered to make the model relatable to real-world phenomena. Often, finding stability conditions for numerical solutions becomes difficult, so an unconditionally stable solution is more appreciable. The homotopy analysis method (HAM), a method known for its unconditional stability, has been used to solve a three-dimensional mathematical model (ADE) along with its initial conditions. Man-made sources show more impact than equal-strength natural sources in the aquifer-aquitard system.

Mathematical Analysis on Dispersion Relation of Rayleigh Wave for an Initially Stressed Magneto-Poroelastic Material with Corrugated and Impedance Boundary

This study showcases the Mathematical modelling of elastic surface waves and the analysis of dispersion relations of Rayleigh-type of wave in an initially stressed homogeneous porous magneto-elastic material having corrugated and impedance boundary exhibitions. Adoption of the classical harmonic method of wave analysis, non-dimensionalization of the resulting equations of motion, and corrugated-impedance boundary conditions produced by the modeled problem are also captured. The dispersion equation was analytically and graphically presented. Effects of the contributing physical quantities, such as impedance and corrugated boundary parameters, on Rayleigh waves for the chosen material are analyzed. The magnetic field’s influences and the wavenumber associated with the corrugated boundary surfaces on the material increase the dispersion relations of the Rayleigh wave profile on the material. In addition, we noted that the dispersion relations of the wave increases for increasing phase velocity and some parameters of voids. Also, a void parameter triggered a decrease on the dispersion profile of the Rayleigh wave when increased while noting some uniform exhibitions from other voids coefficients. This work holds the potential to provide more insights into studies involving displacement distributions in earthquake sciences, stress-strain analysis in Structural Engineering materials, among others.

THE NARROW-BAND FILTER BASED ON A MAGNETOPHOTONIC CRYSTAL INVOLVING LAYERS WITH HYPERBOLIC DISPERSION LAWS

Subject and Purpose. Narrow-band filters are among the basic components of modern communication systems, instruments for spectros- copy, high-sensitivity sensors, etc. Photonic crystal structures open up broad possibilities for creating compact-sized, narrow-band filters in the optical and terahertz ranges. Tuning of spectral characteristics of photonic crystal filters is usually carried out through introduction of certain elements into their structure that are sensitive to external electric and magnetic fields. This work has been aimed at investigating electrodynamic characteristics of one-dimensional magnetophotonic crystals with structural layers characterized by "hyperbolic" disper- sion, and suggesting a multichannel, narrow-band filter on their base. Methods and Methodology. The dispersion equation for excitations in an infinite magnetophotonic crystal has been obtained within the framework of the Floquet-Bloch theory, with the use of fundamental solutions of Hill’s equation. The transfer matrix approach has been used to obtain an analytical expression for the transmission coefficient. Results. The band diagram of the one-dimensional magnetophotonic crystal has been analyzed for the case where one of the layers on the structure’s spatial period is characterized by a hyperbolic dispersion law. The areas of existence of surface wave regimes have been found for such layers for the case of normal incidence of the wave upon the finite-seized magnetophotonic crystal. Frequency dependences of the transmission coefficient are characterized by a set of high-Q resonant peaks relating to Fabry-Perot resonances in a periodic struc- ture of finite length. Conclusions. Application of a finite-seized, one-dimensional magnetophotonic crystal is considered as of a means forachieving mul- tichannel optical filtering and formation of a frequency comb. Expressions for the dispersion equation and transmission coefficient have been obtained within the framework of the Floquet-Bloch theory and with the use of the transfer matrix. The feasibility of surface mode excitation has been shown for gyrotropic layers of the periodic structure characterized by a hyperbolic dispersion law, for the case of nor- mal incidence upon the magnetophotonic crystal. The spectral response of the filter contains narrow-band peaks with a high transmission efficiency. By increasing the number of the structure’s periods it is possible to form a frequency comb, which effect can be useful for appli- cations in metrology and modern optical communication systems.

Solitary waves for dispersive equations with Coifman–Meyer nonlinearities

Solitary waves for dispersive equations with Coifman–Meyer nonlinearities

Unconditional well-posedness for the periodic Boussinesq and Kawahara equations

<abstract><p>In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.</p></abstract>

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

Observability of dispersive equations from line segments on the torus

Observability of dispersive equations from line segments on the torus

The uniform spreading speed in cooperative systems with non-uniform initial data

This paper considers the spreading speed of cooperative nonlocal dispersal systems with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which decreasingly depends only on the smallest decay rate of initial data. The decreasing property of the uniform spreading speed in the smallest decay rate further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components.

High-order cavity coupled plasmon polaritons in a cavity-monolayer MoS<sub>2</sub> system

Compared to graphene, two-dimensional (2D) transition metal sulfides, represented by mono-/few-layer MoS<sub>2</sub>, have tunable non-zero bandgap, which make their application in optoelectronic devices more advantageous. By using classical electromagnetic theory and finite element method (FEM), we investigate the cavity coupled plasmon polaritons (CCPPs) formed through the coupling between cavity modes in a resonator and plasmons in monolayer MoS<sub>2</sub>, particularly calculate and verify the properties of the high-order CCPPs. In previous work, it was demonstrated that the substrates, defects, and polycrystalline grains of the CVD grown monolayer MoS<sub>2</sub> usually induce weak electron localization, which leads to the deviation from the Drude model based on the approximation of free electron gas. Therefore, here we use the Drude-Smith model with characteristic parameters obtained experimentally to describe the optical conductivity of monolayer MoS<sub>2</sub> in our theoretical calculation and simulation. Then, we not only derive and solve the dispersion equations of the high-order CCPPs, but also verify the existence and analyze the properties of these high-order modes. Specifically, there are three types of CCPPs in the asymmetric cavity-monolayer MoS<sub>2</sub> system, i. e., the FP-like-modes (FPLMs), the surface-plasmon-like modes (SPLMs), and the quasi-localized modes (QLMs). Among them, the FPLMs and QLMs can support high-order modes whereas the SPLM only support the fundamental mode. Based on our model, we calculate the wave localization properties for the 7th-order and 8th-order FPLMs, the 3rd-order and 6th-order QLMs, and the SPLM. These theoretical results are in good agreement with the simulation results. Moreover, the effects of weak electron localization are also shown by comparing the field distributions of the CCPPs based on the Drude model and Drude-Smith model. It is found that weak electron localization can reduce the coupling between the cavity modes and the plasmons in monolayer MoS<sub>2</sub>. These results can deepen our understanding of the excitation of plasmons in 2D materials as well as the modulation of their properties. Furthermore, the theoretical model can also be extended to other plasmonic systems associated with low-dimensional and topological quantum materials.

Wave propagation analysis in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory

The purpose of this research is to study the propagation of surface waves in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory. A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory, and classical theory. It has also been concluded that there exist cut-off frequency for the wave propagating in size-dependent materials based on the nonlocal strain gradient theory. The dispersion equation which surface wave speed satisfies is derived from the free traction condition on the surface of half-space with consideration of electrically open circuit conditions. The effect of the nonlocal parameter, the strain gradient parameter on the existence of surface waves as well as the Rayleigh wave propagation is illustrated through some numerical examples.

A Brief Historical Overview and Mathematical Equations of Soliton

bstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that maintains its shape even after colliding with other similar localised wave packets and propagates freely at a constant speed. The medium's balanced cancellation of nonlinear and dispersive effects is responsible for its extraordinary stability. Dispersive effects are a characteristics of several systems where a wave frequency determines how fast it travels. Subsequently, it was shown that soliton solutions offer stable solutions for a broad range of weakly nonlinear dispersive partial differential equations that characterise physical systems. Solitons have a self-restoring quality that makes them desirable for long-distance, high-speed transmissions. The strain on networks is quickly reaching a critical threshold as transmission speeds over longer distances are now approaching 40 Gbit/sec. In the future, solitons might offer intriguing alternatives.

The Study on the Theoretical Perspectives of Soliton and Its Applications

Abstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that propagates freely at a constant speed and keeps its shape even after colliding with other similar localised wave packets. Its exceptional stability arises from the balanced cancellation of nonlinear and dispersive effects in the medium. In engineering, a soliton wave is a self-reinforcing solitary wave packet that keeps its shape as it travels at a steady speed. It results from the medium's non-linear and dispersive effects cancelling each other out. Many systems exhibit dispersive effects, in which the wave frequency controls the wave's speed of propagation. It was then demonstrated that soliton solutions provide stable solutions for a wide class of dispersive partial differential equations that are weakly nonlinear and describe physical systems. Solitons are preferred for high-speed, longdistance transmissions because of their self-restoring characteristics. Given that transmission speeds over longer distances are now close to 40 Gbit per second, the strain on networks is rapidly approaching a critical threshold. Solitons may provide interesting alternatives in the future. Non-linear wave equations that describe how waves propagate in specific physical systems are solved to get soliton solutions. These waves appear as solutions in mathematical models of various systems, such as optical wave-guides, crystal lattice vibrations, and water waves.

Reflection of Elastic Waves in Dipolar Gradient Half-Space under the Control of External Magnetic Field

This paper investigated the reflection of plane waves at the interface of dipolar gradient elastic solids under the control of an external magnetic field. This study focused on the increasing influence of the microstructural effect as the incident wavelength approaches the characteristic length of the microstructure or at higher frequencies. Initially, the dispersion equation for the propagation of elastic waves was derived from the dipole strain gradient theory and Maxwell’s electromagnetic theory. Subsequently, the amplitude ratios of various reflected waves to incident P-waves and incident SV-waves were calculated based on the interface conditions. Finally, the numerical results were used to discuss the impact of the external magnetic field and microstructural characteristic length on the propagation of the reflected wave. It was observed that the microstructural effect generated new wave modes and introduced dispersion characteristics into the elastic waves. Conversely, the external magnetic field primarily influences the amplitude of the elastic wave propagation via the Lorentz force without creating new wave modes or affecting the dispersion properties of the elastic wave in the dipolar gradient elastic solid.

Formation Model for Acoustic Characteristics of Solid Media with Ordered Fracturing

Introduction. The development of new structural materials and improvement of existing technologies for the production of new products on their basis lead to the emergence of new types of medium discontinuities. Therefore, the development of new models of discontinuities that take the previously ignored parameters into account seems to be relevant for the purposes of nondestructive testing and structural measurements. This concerns, e.g., the roughness of adjacent surfaces of microcrack ordered sets.Aim. Theoretical substantiation for the processes of elastic waves propagation through an elastic medium containing an ordered lattice of microcracks with boundary conditions in the linear slip approximation, modified by taking into account the parameters of micro-convexities of microcrack rough boundaries. Database formation for experimental studies aimed at determining the physical and mechanical characteristics of structural materials.Materials and methods. The acoustic characteristics of materials were determined based on the derivation and solutions of dispersion equations describing the formation and propagation of effective longitudinal, transverse, and Rayleigh surface elastic waves in elastic media with ordered cracking. Their values were also used to determine the effective speed of Rayleigh surface waves.Results. The conducted simulation of elastic wave formation processes showed that an increase in the concentration of microcracks leads to a decrease in the phase velocities of effective longitudinal, transverse, and surface waves, as well as to an increase in the attenuation coefficients at given ultrasound frequencies and material parameters.Conclusion. The radius of the microsphere that replaces the surface micro-convexity and the roughness parameter Rz have a significant impact on the formation of physical and mechanical characteristics of materials, which are determined by the results of ultrasonic measurements. The developed model can be recommended as a basis for interpreting the results of ultrasonic measurements.

О резонансах академика И.И. Воровича в контактных задачах с деформируемым штампом в сейсмологии

In this paper, for the first time, methods for obtaining resonant frequencies in plane and spatial contact prob-lems for deformable stamps are presented in a complex. Their existence was predicted by academician I.I. Vorovich. He constructed one-dimensional models of de-formable stamps consisting of a rigid stamp and a spring, illustrating the occurrence of resonances. Currently, thanks to the use of block element methods, it has become possible to investigate real contact problems with a deformable stamp. Earlier in the works of the authors, the existence of a new type of earthquakes, called starting, was established. In the course of the study, some functionals remained uncomputed, the role of which was not clear. It is shown that they serve to construct dispersion curves giving resonant frequencies predicted by academician I.I. Vorovich. A study is carried out for different types of contact problems and shows how the method of constructing dispersion equations becomes more complicated, both due to the complication of the shape of stamps and the complication of the rheologies of deformable stamps. Semi-infinite and deformable stamps of simple rheology in the form of a quarter plane are considered. In addition, a semi-infinite stamp with the rheology of Kirchhoff plates is analyzed. The obtained results provide an answer to one of the methods of constructing dispersion equations for I.I.Vorovich resonances in contact problems for different deformable stamps. The study is closely related to the identification of new precursors of seismicity, as well as to solve some problems in the theory of strength.

An investigation of torsional surface wave in a piezoelectric fiber-reinforced composite layer imperfectly bonded to a functionally graded half-space

This article presents a comprehensive study of torsional surface wave propagation in a piezoelectric fiber-reinforced composite (PFRC) layer bonded to functionally graded (FG) half-space considering electrically short and open boundary conditions. The Strength of Materials (SM) and Rule of Mixtures (ROM) approaches have been adopted to determine the effective properties of the PFRC. The non-ideal mechanical and electrical interfacial boundary has been considered for the problem which appears in various realistic scenarios. For the torsional surface wave propagation through PFRC bonded to FG half-space, the dispersion equation has been derived analytically using Whittaker function. The obtained results are validated and matched with existing results by considering special cases. The variation of phase velocity due to the influence of fiber volume fraction, interfacial imperfection parameters and functionally grading parameters have been analyzed and illustrated by means of graphs. Furthermore, the distribution of the field variables, stress and electric displacements across the composite are presented through surface plots. The model holds potential applications in designing and manufacturing transducers, actuators and sensors. The use of PFRC can be found in a wide range of cutting-edge technology, including vibration-canceling gadgets, energy harvesters, and pressure sensors.

Spatial distribution and modelling of 239+240Pu in the sediments and seawater columns of the South China Sea and Indian Ocean

In order to investigate the 239+240Pu potential influence in the ocean, and develop a new method for rapidly monitoring radioactive pollution, the 239+240Pu spatial distribution in the South China Sea (SCS) and the Indian Ocean (IND) sediments is analyzed by SF-ICP-MS (ELEMENT 2). The inventory-weighted mean activities of 239+240Pu were 0.413 ± 0.333 mBq/g, 0.128 ± 0.044 mBq/g, and 0.483 ± 0.606 mBq/g in the sediments of the SCS, eastern IND, and Arabian Sea, respectively. The 239+240Pu activity spatial distribution in the SCS sediments was influenced by the current, the vertical distribution of Pu in seawater, and the transport of particulate matter. The 239+240Pu activity spatial distribution in the IND sediments could be impacted by Antarctic Intermediate Water. The average of 240Pu/239Pu atomic ratios were 0.258 ± 0.034, 0.219 ± 0.031, and 0.212 ± 0.028 in the sediments of the SCS, eastern IND, and Arabian Sea, respectively. The 240Pu/239Pu atomic ratios in the SCS and IND indicate that Pu from the Pacific Proving Ground (PPG) is transported to the IND via the SCS internal current and transverse ocean currents within Indonesia. In addition, a seawater advection-dispersion equation (S-ADE) model is established based on the actual physical processes of radionuclides in the seawater column and well fitting results were obtained (R2 = 0.49 to 0.99). The 239+240Pu data and the geographic information from the sample site were used to correct the Pu distribution in the seawater. The calculated 239+240Pu mean concentrations in the surface seawater were 2.465 mBq/m3 and 2.205 mBq/m3 for the SCS and the eastern IND seawater, respectively, and the result is consistent with the previous measurements. Then, the 239+240Pu stored in the study area of SCS and eastern IND was estimated to be approximately 1.0–1.4% of the global ocean based on the model. This study provides a useful model for guiding and designing future monitoring of pollution by anthropogenic Pu and other isotopes.

Dispersion in Shallow Moment Equations

Shallow moment models are extensions of the hyperbolic shallow water equations. They admit variations in the vertical profile of the horizontal velocity. This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables. The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction. Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface. The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given. A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.

Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

AbstractThe asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.