Direct Algebraic Approach Research Articles

Investigation of optical soliton solutions for the cubic-quartic derivative nonlinear Schrödinger equation using advanced integration techniques

This paper aims to investigate optical soliton solutions in the context of the cubic-quartic derivative nonlinear Schrödinger equation with differential group delay, incorporating perturbation terms for the first time. Motivated by the need to better understand soliton dynamics in advanced optical communication systems, we employ three integration techniques: the direct algebraic approach, Kudryashov’s method with an addendum, and the unified Riccati equation expansion method. Our study reveals that, by appropriately selecting parameter values, the resulting solutions include Jacobi elliptic functions that describe straddle solitons, bright, dark, and singular solitons. We also identify the conditions under which these soliton pulses can exist. Furthermore, we provide numerical simulations to illustrate these solutions under specific parameter settings, highlighting their potential applications in optical fiber systems.

Investigating dispersive optical solitons with the generalized stochastic perturbed Schrödinger–Hirota equation incorporating power-law nonlinearity and multiplicative white noise

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties.

Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences

In this paper, the extended direct algebraic approach with the general fractional derivative is employed to attain various novel wave structures of the non-linear fractional Tzitzéica type non-linear evolution equations in the form of kink, singular, periodic singular, dark, bright and dark-bright combine solitons. For the purpose to illustrate the physical behavior of the acquired solutions, some of the extracted results are sketched in the pattern of 3-D and 2-D plots which show the efficiency and authenticity of the proposed method. The under consideration method can also utilize to any other non-linear model appears in optics and engineering.

Propagation of solitary wave in micro-crystalline materials

In this article, the propagation of waves in micro-crystalline materials is governed by the structure of the strain wave equation and takes into consideration various dimensions of micro-crystalline. Micro-crystalline materials that are deserving of special attention in material physics. The strain wave equation represents the dynamic behavior associated with multiple phenomena of a physical nature. The new extended direct algebraic methodology is applied to acquire the different types of exact solitonic solutions. This technique stands out as one of the most effective approaches for producing a diverse set of exact solutions to nonlinear partial differential equations. By applying a new extended direct algebraic approach, we get solutions in the form of smooth periodic, anti-dark, anti-bell-shape, periodic bright, Combined bright-dark soliton, mixed-periodic solution, anti-kink formations, Stumpons, mixed periodic solitons, and decaying cusped solitons. Furthermore, two-dimensional, three-dimensional, and contour plots are created for different solutions, helping us make sense of their physical significance more clearly. The importance of the obtained results lies in their ability to represent diverse and complex phenomena in mathematical and physical systems.

On the dynamics of soliton solutions for the nonlinear fractional dynamical system: Application in ultrasound imaging

This article examines the third-order non-linear fractional Westervelt model by using a newly developed integration tool termed the new extended direct algebraic approach. The different wave structures to the considered model are secured in various forms, including bright, dark, and combo solitons. The application of wave structures is advantageous in the examination of sound wave propagation and high amplitude phenomena in the fields of medical imaging and therapy. These solutions are effective in facilitating ultrasound propagation in tissue, underwater acoustics, acoustic cavitation, acoustic levitation, and other related applications. The internal tissue of human beings can be visualized and studied through the use of ultrasound imaging technologies in the field of medicine. This technology possesses numerous uses in both industrial and medicinal sectors. There has been a growing interest in fractional nonlinear partial differential equations due to their ability to describe various complex occurrences and exhibit more dynamic architectures of localized wave solutions. On employing accurate parameter values, multiple graphical representations are generated to provide the visual depiction of the acquired results. The results of this study indicate that the chosen methodology effectively improves nonlinear dynamical processes. The findings indicate that the selected methodology demonstrates efficacy, feasibility, and versatility when applied to intricate systems across several domains, with a special emphasis on ultrasonic imaging. The findings indicate that the system exhibits a potentially significant abundance of soliton structures.

Optical soliton solutions of the coupled Radhakrishnan-Kundu-Lakshmanan equation by using the extended direct algebraic approach

The analytical soliton solutions place a lot of value on birefringent fibres. The major goal of this study is to generate novel forms of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation, which depicts unstable optical solitons that arise from optical propagations using birefringent fibres. The (presumably new) extended direct algebraic (EDA) technique is used here to extract a large number of solutions for RKLE. It gives soliton solutions up to thirty-seven, which essentially correspond to all soliton families. This method's ability to determine many sorts of solutions through a single process is one of its key advantages. Additionally, it is simple to infer that the technique employed in this study is really straightforward yet one of the quite effective approaches to solving nonlinear partial differential equations so, this novel extended direct algebraic (EDA) technique may be regarded as a comprehensive procedure. The resulting solutions are found to be hyperbolic, periodic, trigonometric, bright and dark, combined bright-dark, and W-shaped soliton, and these solutions are visually represented by means of 2D, 3D, and density plots. The present study can be extended to investigate several other nonlinear systems to understand the physical insights of the optical propagations through birefringent fibre.

Traveling waves solutions of Hirota–Ramani equation by modified extended direct algebraic method and new extended direct algebraic method

In this paper, new exact traveling wave solutions are obtained by Hirota–Ramani equation. The many exact complex solutions of several types of nonlinear partial differential equations (NPDEs) are presented using the modified extended direct algebraic approach and new extended direct algebraic method, which is among the most effective mathematical techniques for finding a precise solution to NPDEs and put into a framework of algebraic computation. By selecting different bright and solitary soliton forms and by creating various analytical optical soliton solutions for the investigated equation, we hope to demonstrate how the analyzed model’s parameter impacts soliton behavior. It is possible to obtain new, complex solutions for nonlinear equations like the [Formula: see text]-dimensional Hirota–Ramani equation.

Novel soliton wave solutions of a special model of the nonlinear Schrödinger equations with mixed derivatives

In this study, the influence of the integrability requirement on nonlinear Schrödinger equations with mixed derivatives is examined. The Rangwala–Rao (RR) equation is named after A. Rangwala, who in 1990 was the first to quantify these effects in their entirety. The aim of our research is identifying how to generate individual waves and how they interact with one another. This is to get a better understanding of the dispersion effect and the progressive variation of the electric field envelope during pulse propagation in optical fibers. The direct algebraic method (DAM), the extended direct algebraic approach (exDAA), and the extended Riccati expansion technique (exREM) are used to develop unique solitary wave solutions for the model under consideration. These numerically calculated solutions illustrate the dynamic behavior of an optical pulse traveling through a cable. Comparing our results to those of other researchers demonstrates the originality of our work.

Ion-acoustic wave dynamics and sensitivity study in a magnetized Auroral phase plasma

The dynamic behavior of ion-acoustic waves (IAWs) inside a magnetized auroral zone plasma including ions as well as two-temperature electrons is investigated. The Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are provided using the reductive perturbation method (RPT). In fluid dynamics, plasma physics, and electrodynamics, mKP equation can be utilized to characterize wave phenomena. It increases our interest due to its significant physical and mechanical value. The new extended direct algebraic approach is used to compute several traveling wave patterns for the nonlinear mKP equation. It is a simple, concise and general method in symbolic computations for solitary wave solutions for nonlinear problems. It yields dark solitary solutions, the community for semi-dark soliton results, dark singular soliton effects, kink soliton results, and soliton consequences encompassing trigonometric function, mixture hyperbolic function, as well as rational functions. Some arbitrarily chosen solutions are displayed visually for physical understanding. Moreover, the sensitivity of the perturbed dynamical structure equivalent to the mKP equation has been explored. Furthermore, the effect of IAWs spectral index on the parameters of the solitary-wave solutions and the sensitivity of the perturbed dynamical system of mKP equations is investigated in detail.

The First Integral of the Dissipative Nonlinear Schrödinger Equation with Nucci’s Direct Method and Explicit Wave Profile Formation

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1+1)− dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci’s scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci’s direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci’s scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution.

A study of phase portraits, multistability and velocity profile of magneto-hydrodynamic Jeffery–Hamel flow nanofluid

Bifurcation analysis is studied for the first time in magneto-hydrodynamic (MHD) Jeffery–Hamel flow nanofluids. The dimensionless ordinary differential equations is formulated by using the similarity transformations. The phase portraits of the travelling wave solutions are examined using bifurcation theory on the dynamical system of the considered equation. Moreover, multistability and sensitivity analysis is used to examine quasi-periodic and periodic behavior for various initial conditions. The (MHD) Jeffery–Hamel flow solutions are derived analytically by applying the extended direct algebraic approach and the influence of the different physical parameters for velocity profiles, have been examined. It is found that the physical parameter i.e, Reynolds number and channel angle rising in convergent flow results in an increase in the velocity profiles, signifying that backflow is excluded, while an inverse behavior can be attributed to a diverging flow. • Magneto-hydrodynamic Jeffery–Hamel flow has been observed by bifurcation analysis. • All possible phase portraits are drawn. • The influence of the magnetic field and nanoparticles on the magneto-hydrodynamic Jeffery–Hamel flow has been studied. • The influence of the different physical parameters on the velocity profiles have been examined.

Bifurcation, chaotic and multistability analysis of the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation with external perturbation

This paper presents a theory of weakly nonlinear wave propagation in superposed fluids in the existence of magnetic fields. The extended direct algebraic approach has been utilized to draw and examine the solitonic wave solutions to the -dimensional elliptic nonlinear Schrödinger equation (ENLSE). These obtained solutions are more novel and valuable to researchers for understanding physical marvels. The graphical representation of some selected exact solutions is reported with different parametric values to show their propagation. Furthermore, the examined equation is changed into the planer dynamical structure by utilizing Galilean transformation. By utilizing the idea of bifurcation, all the possible phase portraits of nonlinear and super nonlinear traveling wave solutions have been explored. Moreover, after adding an external force to the dynamical system, the periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D and time series are also observed. Meantime, for various initial conditions, the sensitivity of the derived solutions is investigated in detail. It is discovered that the considered equation is multistable for the same values of parameters but different initial conditions.

Soliton solutions, Lie symmetry analysis and conservation laws of ionic waves traveling through microtubules in live cells

This study explores a fourth-order nonlinear symmetric solution to ionic waves in living microtubules. Lie group analysis and the new extended direct algebraic approach are used to build solitary wave solutions. We reduce the partial differential equation based on symmetry into ordinary differential equations. Solved equations yield novel single wave patterns. For physical explanation, specific solitary wave structures are visually shown. The solutions include the anti-kink, kink shape, the absolute value of the spike shape, singular kink wave shape, and dark-singular soliton solution. Utilizing the multiplier technique, we establish the equation’s local conservation laws. Some solution sketches depict the model’s vision.

APPRAISAL OF ANALYTICAL SOLUTIONS FOR (2 + 1)-DIMENSIONAL NONLINEAR CHIRAL SCHRÖDINGER EQUATION

The solitary wave solutions of the [Formula: see text]-dimensional nonlinear Chiral Schrödinger ([Formula: see text]-D CNLS) equation by making use of the modern extended direct algebraic approach are investigated in this paper. Such a nonlinear model’s different categories of solitary waves solutions are found. The division of semi-dark solitons, singular dark-pitch solitons, single solitons of type 1 along with 2, intermixed hyperbolically, trigonometrically and rational solitons are developed and evaluated using software tools for numerical calculations. Detailed graphical interpretation of results with deliberation is also discussed. This paper effectively depicts 3D- and 2D-contour plots to provide further detail about the physical characteristics of the obtained solutions. We compare our solutions to previously reported findings along with study motivation and future directions. The fractional quantum hall effect’s edge states are defined by the discussed model.

Diverse acoustic wave propagation to confirmable time–space fractional KP equation arising in dusty plasma

In this study, the (3+1)-dimensional fractional time–space Kadomtsev–Petviashivili (FTSKP) equation is considered and analyzed analytically, which propagates the acoustic waves in an unmagnetized dusty plasma. The fractional derivatives are studied in a confirmable sense. The new modified extended direct algebraic (MEDA) approach is adopted to investigate the diverse nonlinear wave structures. A variety of new families of hyperbolic and trigonometric solutions are obtained in single and different combinations. The obtained results are also constructed graphically with the different parametric choices.

Analysis of electron acoustic waves interaction in the presence of homogeneous unmagnetized collision-free plasma

In this research, the transmission and interaction of nonlinear electron acoustic waves (EAWs) in such an unmagnetized, homogeneous, collision-free plasma composed of hot and cold electrons together with stationary ions throughout in the background have been analyzed. For the small-amplitude limit, the Korteweg–de Vries (KdV) equation for (EAWs) have been extracted. For electron acoustic solitary waves (EASWs), using the new extended direct algebraic approach, soliton solutions have also documented. The parametric analysis demonstrated that the hot to cold electron ratio and hot electron superthermal play a key role in changing the (EASWs) amplitude. The family of semi-bright solitons, dark singular solitons, Type 1 as well as 2 single solitons, trigonometric, intermingled hyperbolic and rational solitons was constructed and tested with the assistance of the innovative package software of numerical computations. The results show that the method is clear and efficient, produces analytical results in a generalized form, and these findings can also help resolve the difficulties and predicaments in the relevant disciplines of plasma physics and may be useful for studying the relationship between two (EASWs) in astrophysical and laboratory plasma. The solutions presented in this prototype are the latest in a literature review. For physical interpretation, some randomly selected solutions are shown graphically. Conclusions are held at the end.

New general extended direct algebraic approach for optical solitons of Biswas-Arshed equation through birefringent fibers

In this paper, the new general extended direct algebraic method is applied to achieve optical solitons of Biswas-Arshed equation with birefringent fiber involving two component vector soliton. Family of semi-bright soliton, dark singular soliton, singular soliton of Type 1 as well as 2, trigonometric, mixed hyperbolic and rational solitons are constructed and also verified by the aid of modern technical computing package program. The observed findings demonstrate that the approach is straightforward and easy, offers analytical results in generalized manner also these findings may help to address the challenges and predicaments that exist in the optical fibers and telecommunications sector. The solutions encountered with this model are completely latest to the academic literature. Some randomly chosen solutions are illustrated graphically for the physical interpretation. At the end, conclusions are announced.

New optical soliton solutions for Triki–Biswas model by new extended direct algebraic method

The study focuses on the use of a direct algebraic approach to the analysis of the Triki–Biswas (TB) model. This model addresses the distribution of ultrashort pulses in optical fiber in the presence of non-Kerr dispersion concept and group velocity dispersion. However, using the new extended direct algebraic method, we have obtained various optical soliton solutions for the TB model. The optical soliton solutions are new and reliable compared to the existing methods.

An NOx Sensor-Based Direct Algebraic Approach-Newton Observer for Urea Selective Catalytic Reduction System State Estimations

NOx sensor-based state estimations for urea-based selective catalytic reduction (SCR) systems have attracted much attention in the past several years because of their significant importance in achieving high NOx conversion efficiency and low ammonia slip at low operation cost. Most of the existing SCR state estimation techniques require sophisticated design processes and significant tuning efforts, which may prevent them from widespread applications to production urea-SCR systems. In addition, the existing SCR state observers may not be able to achieve fast and accurate estimations due to the corresponding slow estimation error dynamics. The purpose of this study was to design a straightforward and effective NOx sensor-based SCR state estimation algorithm for decoupling post-SCR NOx sensor signals (NOx concentration, ammonia concentration), and for estimating ammonia coverage ratio of the urea-SCR systems. A singular-perturbation-based approach was applied to attain the reduced-order SCR model by decoupling the fast NO and NH3 concentration dynamic models from the slow ammonia coverage ratio dynamics model. Based on the reduced-order model, a direct algebraic approach (DAA)-Newton observer was proposed for estimating ammonia coverage ratio. The achieved ammonia coverage ratio estimation was applied to estimate the post-SCR NOx and NH3 concentrations. Simulation verification results under US06 cycle proved the effectiveness of the proposed method in accurately estimating the aforementioned key SCR states. The proposed observer can potentially be popularly applied to the production SCR systems for the advanced SCR control systems and on-board diagnostics.

Modelling mortality of a stored grain insect pest with fumigation: Probit, logistic or Cauchy model?

Computer simulation models can provide a relatively fast, safe and inexpensive means to judge and weigh the merits of various pest control management options. However, the usefulness of such simulation models relies on the accurate estimation of important model parameters, such as the pest mortality under different treatments and conditions. Recently, an individual-based simulation model of population dynamics and resistance evolution has been developed for the stored grain insect pest Rhyzopertha dominica, based on experimental results showing that alleles at two different loci are involved in resistance to the grain fumigant phosphine. In this paper, we describe how we used three generalized linear models, probit, logistic and Cauchy models, each employing two- and four-parameter sub-models, to fit experimental data sets for five genotypes for which detailed mortality data was already available. Instead of the usual statistical iterative maximum likelihood estimation, a direct algebraic approach, generalized inverse matrix technique, was used to estimate the mortality model parameters. As this technique needs to perturb the observed mortality proportions if the proportions include 0 or 1, a golden section search approach was used to find the optimal perturbation in terms of minimum least squares (L2) error. The results show that the estimates using the probit model were the most accurate in terms of L2 errors between observed and predicted mortality values. These errors with the probit model ranged from 0.049% to 5.3%, from 0.381% to 8.1% with the logistic model and from 8.3% to 48.2% with the Cauchy model. Meanwhile, the generalized inverse matrix technique achieved similar results to the maximum likelihood estimation ones, but is less time consuming and computationally demanding. We also describe how we constructed a two-parameter model to estimate the mortalities for each of the remaining four genotypes based on realistic genetic assumptions.