Hyperbolic Tangent Method Research Articles

Shocks in an electronegative plasma with Boltzmann negative ions and κ-distributed trapped electrons

We derived a Schamel-Burgers equation to study the dynamics of nonlinear structures in dissipative electronegative plasma having Boltzmann negative ions and κ-distributed trapped electrons. A recently introduced Tangent hyperbolic method has been employed to get the solutions of the differential equations which contain fractional nonlinearity. The effects of different physical parameters particularly, the kinematic viscosity, the superthermality, the trapping efficiency and the electronegativity factor on the ion acoustic (IA) shock profiles have been elaborated. In case of non-dissipative electronegative plasma, the small amplitude double layers (DLs) have also been investigated. It has been elaborated that the DLs strongly depend on the system parameters. The results illustrate that the superthermality index and trapping parameter play disruptive role in the formation of DLs. Likewise, the electronegativity factor plays a dominant role in the shaping of the compressive DLs. The study can be supportive to understand the behavior of nonlinear structures as observed in the nature and laboratory plasmas.

Mathematical analysis of a tumour-immune interaction model: A moving boundary problem

A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level.

Fractional effects on solitons in a 1D array of rectangular ferroelectric nanoparticles

ABSTRACT In this paper, we present non-local effects on solitons in a one-dimensional (1D) array of ferroelectric nanoparticles. Using the discrete tangent hyperbolic method, several fractional function solutions are obtained such as fractional hyperbolic, trigonometric and rational function solutions. As results, we derive many solitons such as dark, bright and rogue waves. We show that the fractional parameter α has an effect on the width and on the amplitude of the above-obtained solitons. Their width and their amplitude are modified by the value of the fractional parameter α. Fractional derivative has a memory effect on ferroelectric nanoparticles. Applications of the obtained results are also presented.

Study of obliquely propagating electron acoustic shock waves with non-extensive electron population

Obliquely propagating electron acoustic shock waves in magnetized plasma composed of stationary ions, cold and non-extensive hot electrons are investigated by deriving Korteweg–de Vries Burgers (KdVB) equation. The tangent hyperbolic method is used to solve the KdVB equation in dissipative medium. The dissipation effect is introduced in the model by means of kinematic viscosity term. The analytical calculations of the KdVB equation shows that the structures (amplitude, velocity and width) of the shock waves are modified significantly with kinematic viscosity (η0), obliqueness (kz) and magnetic field (ωc). Since plasmas are ubiquitously permeated with magnetic field, it is pertinent to explore the characteristics of KdVB equation in a magnetized plasmas.

Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method

There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their high non-linearity and the huge number of parameters of which they consist. In this article we determine, using the hyperbolic tangent (tanh) method, travelling wave solutions to non-linear evolution models of interest in biology and physics. These solutions have recognizable properties expected of other solutions and thus can be used to deduce properties of the general solutions.

Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons

The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV–Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.

Nonlinear dynamics of microtubules and series expansion unknown function method

Microtubules (MTs) are the most important part of cytoskeleton. In this paper we deal with two basic nonlinear differential equations coming from the two known models describing nonlinear dynamics of MTs. These equations are solved using the series expansion unknown function method (SEUFM). Trying to recognize the most general mathematical procedure for solving these equations the solutions are compared with those obtained earlier using the tangent hyperbolic function method (THFM) and the simplest equation method (SEM). In all these three approaches we express the solutions of these equations as series expansions. In the cases of THFM and SEM the functions existing in the series are known while SEUFM assumes unknown function.

A model for porcine reproductive and respiratory syndrome with time-dependent infection rate: traveling wave solution

Porcine reproductive and respiratory syndrome (PRRS) gives rise to reproductive disorders in sows and problem with respiratory system in piglets and young pigs. This disease creates serious economic losses to major pork producing countries. The disease, which is characterized by high morbidity and significant mortality, combined with its potential for rapid spread, can devastate the pig industries of the affected countries. However, not much is known about the spatial transmission of PRRSV (porcine reproductive and respiratory syndrome virus) in growing pigs. In previous models, the infection rate has been assumed to be constant with time. Experimental studies on specific cases of this viral infection suggest that this assumption might not hold. A structured model for the spread of PRRSV has therefore been constructed, incorporating time and spatial dimensions as well as the decline of infection rate with time. Using the traveling wave coordinate and the modified extended hyperbolic tangent method, we derive analytical solutions to the model system. Stability and phase plane analyses are also carried out in order to gain insights into the spatial spread of PRRS as time progresses.

Nonlinear dissipative and dispersive electrostatic structures in unmagnetized relativistic electron-ion plasma with warm ions and trapped electrons

In this paper, we have investigated electrostatic solitary and shock waves in an unmagnetized relativistic electron-ion (ei) plasma in the presence of warm ions and trapped electrons. In this regard, we have derived the trapped Korteweg-de Vries Burgers (TKdVB) equation using the small amplitude approximation method, which to the best of our knowledge has not been investigated in plasmas. Since the TKdVB equation involves fractional nonlinearity on account of trapped electrons, we have employed a smartly crafted extension of the tangent hyperbolic method and presented the solution of the TKdVB equation in this paper. The limiting cases of the TKdVB equation yield trapped Burgers (TB) and trapped Korteweg-de Vries (TKdV) equations. We have also presented the solutions of TB and TKdV equations. We have also explored how the plasma parameters affect the propagation characteristics of the nonlinear structures obtained for these modified nonlinear partial differential equations. We hope that the present work will open new vistas of research in the nonlinear plasma theory both in classical and quantum plasmas.

Propagation of envelope bright breather coupled wave modes in Ablowitz–Ladik chains

We investigate a model of a two coupled Ablowitz–Ladik (AL) lattices which admits the integrable AL model, when it is reduced to symmetric states. We compute the discrete modulational instability gain analytically and verified numerically. The play-role of the coupling parameter ϵ which is the back-bone of the two coupled AL chain is analyzed. We perform molecular dynamics (MD) simulations and elucidate the short time as well as the long time instability of AL chain. We demonstrate that the AL chain supports the formation of bright breather coupled wave modes spanning over 36 lattice sites on either side of the bond-center. We explore a kink solitary profile of solutions for the AL chains by invoking the extended tangent hyperbolic function method (ETHF) embedded with symbolic computation. Spectral stability of the exact solutions is determined by the eigenvalues of the discrete spectrum. We demonstrate the onset of chaos that leads the quasi-periodic dynamics of AL chains through the period-doubling bifurcation route to chaos.

Assessing the Impact of Climate Variability on Cropland Productivity in the Canadian Prairies Using Time Series MODIS FAPAR

Cropland productivity is impacted by climate. Knowledge on spatial-temporal patterns of the impacts at the regional scale is extremely important for improving crop management under limiting climatic factors. The aim of this study was to investigate the effects of climate variability on cropland productivity in the Canadian Prairies between 2000 and 2013 based on time series of MODIS (Moderate Resolution Imaging Spectroradiometer) FAPAR (Fraction of Absorbed Photosynthetically Active Radiation) product. Key phenological metrics, including the start (SOS) and end of growing season (EOS), and the cumulative FAPAR (CFAPAR) during the growing season (between SOS and EOS), were extracted and calculated from the FAPAR time series with the Parametric Double Hyperbolic Tangent (PDHT) method. The Mann-Kendall test was employed to assess the trends of cropland productivity and climatic variables, and partial correlation analysis was conducted to explore the potential links between climate variability and cropland productivity. An assessment using crop yield statistical data showed that CFAPAR can be taken as a surrogate of cropland productivity in the Canadian Prairies. Cropland productivity showed an increasing trend in most areas of Canadian Prairies, in general, during the period from 2000 to 2013. Interannual variability in cropland productivity on the Canadian Prairies was influenced positively by rainfall variation and negatively by mean air temperature.

Nonlinear Electron Acoustic Waves in Dissipative Plasma with Superthermal Electrons

The nonlinear properties of small amplitude electron-acoustic ( EA) solitary and shock waves in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and superthermal hot electrons obeying superthermal distribution, and stationary ions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili-Burgers (KP-Brugers) equation. Some solutions of physical interest are obtained. These solutions are related to soliton, monotonic and oscillatory shock waves and their behaviour are shown graphically. The formation of these solutions depends crucially on the value of the Burgers term and the plasma parameters as well. By using the tangent hyperbolic (tanh) method, another interesting type of solution which is a combination between shock and soliton waves is obtained . The topology of phase portrait and potential diagram of the KP-Brugers equation is investigated.The advantage of using this method is that one can predict different classes of the travelling wave solutions according to different phase orbits. The obtained results may be helpful in better understanding of waves propagation in various space plasma environments as well as in inertial confinement fusion laboratory plasmas.

Modified Zakharov–Kuznetsov equation for a non-uniform electron–positron–ion magnetoplasma with kappa-distributed electrons

We investigate the low-frequency (by comparison with the ion Larmor frequency) electrostatic solitary structures in a spatially non-uniform electron–positron–ion (e–p–i) magnetoplasma with non-Maxwellian electrons. A linear dispersion relation for the obliquely propagating ion acoustic drift wave is derived and it is shown that the non-Maxwellian electron population modifies the dispersion characteristics of the wave under consideration. We also carry out a nonlinear analysis and derive the modified Zakharov–Kuznetsov (MZK) equation for the coupled drift acoustic wave in a non-uniform magnetized plasma. We highlight the differences between the MZK equation and its homogeneous counterpart. We also find the solution of the MZK equation using the tangent hyperbolic method. It is observed that the electron spectral index ${\it\kappa}$, positron concentration, and propagation angle ${\it\alpha}$ alter the structure of the ion acoustic drift solitary waves. The results obtained in this paper may be beneficial to understanding the propagation characteristics of electrostatic drift solitary structures in the interstellar medium and in laboratory experiments where electron–positron plasmas have recently been created by impinging ultra-intense laser pulses on a solid density target at the Lawrence Livermore National Laboratory (LLNL).

On shape changing solutions of a generalized inhomogeneous Hirota equation

In a recent series of papers, Kavitha et al. [2,3,4] solved three inhomogeneous nonlinear Schrödinger (INLS) integro-differential equation under the influence of a variety of nonlinear inhomogeneities and nonlocal damping by the modified extended tangent hyperbolic function method. They obtained several kinds of exact solitary solutions accompanied by the shape changing property. In this paper, we demonstrate that most of exact solutions derived by them do not satisfy the nonlinear equations and consequently are wrong. Furthermore, we study a generalized Hirota equation with spatially-inhomogenetiy and nonlocal nonlinearity. Its integrability is explored through Painlevé analysis and N-soliton solutions are obtained based on the Hirota bilinear method. Effects of linear inhomogeneity on the profiles and dynamics of solitons are also investigated graphically.

Linear and nonlinear analysis of dust acoustic waves in dissipative space dusty plasmas with trapped ions

The propagation of linear and nonlinear dust acoustic waves in a homogeneous unmagnetized, collisionless and dissipative dusty plasma consisted of extremely massive, micron-sized, negative dust grains has been investigated. The Boltzmann distribution is suggested for electrons whereas vortex-like distribution for ions. In the linear analysis, the dispersion relation is obtained, and the dependence of damping rate of the waves on the carrier wave number \(k\), the dust kinematic viscosity coefficient \(\eta _{d}\) and the ratio of the ions to the electrons temperatures \(\sigma _{i}\) is discussed. In the nonlinear analysis, the modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation is derived via the reductive perturbation method. Bifurcation analysis is discussed for non-dissipative system in the absence of Burgers term. In the case of dissipative system, the tangent hyperbolic method is used to solve mKdV–Burgers equation, and yield the shock wave solution. The obtained results may be helpful in better understanding of waves propagation in the astrophysical plasmas as well as in inertial confinement fusion laboratory plasmas.

Eight-Scale Image Contrast Enhancement Based on Adaptive Inverse Hyperbolic Tangent Algorithm

The Eight-Scale parameter adjustment is a natural extension of Adaptive Inverse Hyperbolic Tangent (AIHT) algorithm. It has long been known that the Human Vision System (HVS) heavily depends on detail and edge in the understanding and perception of scenes. The main goal of this study is to produce a contrast enhancement technique to recover an image from blurring and darkness, and at the same time to improve visual quality. Eight-scale coefficient adjustments can provide a further local refinement in detail under the AIHT algorithm. The proposed Eight-Scale Adaptive Inverse Hyperbolic Tangent (8SAIHT) method uses the sub-band to calculate the local mean and local variance before the AIHT algorithm is applied. This study also shows that this approach is convenient and effective in the enhancement processes for various types of images. The 8SAIHT is also capable of adaptively enhancing the local contrast of the original image while simultaneously extruding more on object details.

The Variable Steering Ratio for Vehicle Steer by Wire System Using Hyperbolic Tangent Method

In conventional steering system, during the parking maneuver, driver required large turned on the steering wheel to move the fornt tyre. Thus, it will increase the driver burden when turned the steering wheel. The feature of variable steering ratio (VSR), help to reduce driver burden. Moreover, it improves the vehicle maneuver at lower and high speed. This paper, proposed a control algorithm of variable steering ratio (VSR) in vehicle SBW system. The concept of hyperbolic tangent is used where it not only improved the maneuverability at lower speed, but also reduces the driver burden on the steering wheel. To investigate the effectiveness of the proposed VSR algorithm, the result is compared with conventional steering system

Shape changing nonlocal molecular deformations in a nematic liquid crystal system

The nature of nonlinear molecular deformations in a homeotropically aligned nematic liquid crystal (NLC) is presented. We start from the basic dynamical equation for the director axis of a NLC with elastic deformation and mapped onto a integro-differential perturbed Nonlinear Schrödinger equation which includes the nonlocal term. By invoking the modified extended tangent hyperbolic function method aided with symbolic computation, we obtain a series of solitary wave solutions. Under the influence of the nonlocality induced by the reorientation nonlinearity due to fluctuations in the molecular orientation, the solitary wave exhibits shape changing property for different choices of parameters. This intriguing property as a result of the relation between the coherence of the solitary deformation and the nonlocality reveals a strong need for a deeper understanding in the theory of self-localization in NLC systems.

Solitons and shocks in dense astrophysical magnetoplasmas with relativistic degenerate electrons and positrons

The linear and nonlinear properties of the ion-acoustic (IA) waves are investigated in a relativistically degenerate magnetoplasma, whose constituents are the electrons, positrons, and ions. The electrons and positrons are assumed to obey the Fermi-Dirac statistics, whereas the cold ions are taken to be inertial and magnetized. In linear analysis, various limiting cases are discussed both analytically and numerically. However, for nonlinear studies, the well-known reductive perturbation technique is employed to derive the Zakharov-Kuznetsov and Zakharov-Kuznetsov Burgers equations in the presence of relativistically degenerate electrons and positrons. Furthermore, with the use of hyperbolic tangent method, the equations are simplified to admit the soliton and shock wave solutions. Numerically, it is shown that the amplitude, width, and phase speed associated with the localized IA solitons and shocks are significantly influenced by the various intrinsic plasma parameters relevant to our model. The present analysis can be useful for understanding the collective processes in dense astrophysical environments like neutron stars, where the electrons and positrons are expected to be relativistic and degenerate.

Collision of electromagnetic solitons in a weak ferromagnetic medium

We investigate the propagation dynamics of electromagnetic soliton in a weak ferromagnet with Dzyaloshinskii–Moriya interaction and uniaxial anisotropy along the propagation of electromagnetic (EM) wave. We perform the analysis in the framework of Landau–Lifshitz (LL) equation governing the spin evolution in ferromagnet and Maxwell (M) equations representing the electromagnetic wave propagation. We invoke the reductive perturbation method, and a nonuniform expansion is carried out on the coupled LLM equations which is reduced to a perturbed derivative nonlinear Schrödinger (pDNLS) equation for a new complex field that govern the spin excitations under the influence of EM wave. The pDNLS equation is systematically solved through the tangent-hyperbolic method and a class of soliton solutions are constructed. Further, the collision of EM wave soliton and the influence of Dzyaloshinkii–Moriya interaction is established via Hirota bilinerization method.