Auxiliary Equation Method Research Articles

One‐Step Leapfrog Locally 1D Theory with Higher‐Order Enhanced Absorption for Fine‐Details Simulation

AbstractBased on a higher‐order concept and one‐step leapfrog formulation, an unconditionally stable leapfrog locally 1D (LOD) perfectly matched layer (PML) is proposed to not only enhance the absorbing performance in terminating an unbounded finite‐difference time‐domain (FDTD) algorithm but also improve the computational efficiency compared with the LOD‐FDTD algorithm. Most precisely, the PML is implemented by the auxiliary differential equations method. By applying the proposed scheme to a 3D electromagnetics simulation, the proposal can utilize the unconditionally stable algorithm, one‐step leapfrog formulation, and higher‐order concept in terms of attenuating low‐frequency propagation waves, enhancing the performance, and improving the computational efficiency. Numerical examples including the magic‐T waveguide model, very large‐scale integration circuits, and the structure of metamaterials are investigated to further demonstrate the efficiency, performance, and unconditional stability. The results help conclude that the proposed scheme can attain better performance and efficiency when the time step far surpasses the Courant–Friedrichs–Lewy limit. Most importantly, by employing the one‐step leapfrog formulation in the LOD algorithm, the memory requirements and simulation duration can be reduced significantly, indicating the improvement of the algorithm compared with the published LOD schemes.

Newly modified unified auxiliary equation method and its applications

This study aims to propose a new modified unified auxiliary equation method for the first time. To illustrate the validity of the new modified method, we consider the dimensionless time-dependent paraxial equation which is investigated previously in Tarla and Yilmazer (2022) using the unified auxiliary equation method. As a result, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. The obtained results are compared with the solutions obtained by the unified auxiliary equation method in Tarla and Yilmazer (2022) in the result and discussion section. We conclude that our findings are novel and distinctive, and we give the comparison in the section on results and discussion. It is demonstrated that the modified unified auxiliary equation method provides a more efficient mathematical technique for solving nonlinear partial differential equations. In addition, we draw 3D profiles and counter plots for some obtained solutions with the assistance of a computer program by giving a specific value for the involved parameters.

Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics

In this paper, we consider a coupled space-time fractional nonlinear Schrödinger type equation which can be used for describing nonrelativistic quantum mechanical behavior. Under the help of the fractional complex transform and the conformable fractional derivative sense, we apply the extended auxiliary equation method, the modified He’s semi-inverse method and the improved Weierstrass elliptic function method to obtain many exact solutions. These exact solutions include the bright, dark, bright-dark, bright-like and kinky-bright optical soliton solutions as well as trigonometric function solutions, the Jacobi elliptic function solutions and the general solutions in form of Weierstrass elliptic function. The behaviors of some obtained solutions are displayed through the 3D graphs for different fractional orders by using Matlab. The obtained results show that these proposed methods are straightforward, efficient and can provide more forms of exact solutions, which are helpful to understand the complex physical meaning of space-time fractional coupled nonlinear Schrödinger equation.

Dispersive Optical Solitons to Stochastic Resonant NLSE with Both Spatio-Temporal and Inter-Modal Dispersions Having Multiplicative White Noise

The current article studies optical solitons solutions for the dimensionless form of the stochastic resonant nonlinear Schrödinger equation (NLSE) with both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD) having multiplicative noise in the itô sense. We will discuss seven laws of nonlinearities, namely, the Kerr law, power law, parabolic law, dual-power law, quadratic–cubic law, polynomial law, and triple-power law. The new auxiliary equation method is investigated. We secure the bright, dark, and singular soliton solutions for the model.

A variety of new soliton structures and various dynamical behaviors of a discrete electrical lattice with nonlinear dispersion via variety of analytical architectures

Nonlinear electrical lattices are powerful experimental tool for studying nonlinear dispersive media and creating opportunities for the realistic modeling of electrical solitons. In this work, we adopt the generalized auxiliary equation methods, the first integral method, and the Möbius transformation approach to extract a variety of traveling and solitary wave solutions of the nonlinear Salerno equation describing the nonlinear discrete electrical lattice with nonlinear dispersion. The kink, antikink, dark, bright, peakons, antipeakons, and periodic wave solutions are all derived. The existence criteria of solitons are described. It has been shown that for the selective values of arbitrary constants in an auxiliary equation, the generalized auxiliary equation method II is reduced to the new ‐model expansion method as well as the new extended auxiliary equation method. The impact of free parameters on the obtained solutions is investigated and graphically depicted using physical descriptions. The acquired results are important for the validity of numerical and experimental results and further understanding of the wave propagation in the nonlinear discrete electrical lattice. In addition, the stability analysis of the obtained solitary wave solutions is studied. Furthermore, using the maple software, all derived solutions were checked by re‐entering them into the considered equation.

New analytical solutions and efficient methodologies for DNA (Double-Chain Model) in mathematical biology

In this paper, the double-chain model for deoxyribonucleic acid (DNA) is studied analytically by taking advantage of two reliable methodologies: the new auxiliary equation method (NAEM) and the expa function method. The deoxyribonucleic acid (DNA) contains the genetic information that creatures require for their survival and reproduction. The used approaches produce exact explicit solutions to the double chain model of DNA in mathematical biology, which are presented in this paper. Many novel solutions such as trigonometric, exponential, singular, and hyperbolic solutions, are obtained by using the aforementioned techniques. Numerical simulations are employed to supplement the obtained solutions, making this research biologically relevant. Eventually, kink wave, nonlinear wave, plane wave-front and doubly soliton profiles of solutions are presented in analysis and discussion section. It is indeed important to note that the solutions produced using these techniques are more generalized and can be more useful to demonstrate the internal interactions of the double chain DNA model.

New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model

This article discusses dynamics of the fractal double-chain deoxyribonucleic acid model. This structure contains two long elastic homogeneous strands that serve as two polynucleotide chains of deoxyribonucleic acid molecules, bounded by an elastic membrane indicating hydrogen bonds between the base pairs of two chains. The semi-inverse variational principle and auxiliary equation method are employed to extricate soliton solutions. The collection of retrieved exact solutions includes bright, dark, periodic, and other solitons. The constraint conditions emerge naturally which ensure the presence of these solutions. Additionally, 2D and 3D graphs showing the impact of fractals on solutions are included. These plots use appropriate parameter values. Furthermore, sensitivity analysis of the considered model is also acknowledged. The outcomes reveal that these techniques are reliable, effective, and applicable to various biological systems.

Stochastic optical solitons with multiplicative white noise via Itô calculus

Purpose:This study has been carried out to examine the stochastic optical soliton solutions of the nonlinear Schrödinger’s equation (NLSE) with Kerr law nonlinearity by multiplicative noise in Itô sense and the behavioral changes on soliton dynamics. Methodology:In order to see the noise effect better, bright and dark soliton shapes belonging to the NLSE have been obtained using the subversion of the new extended auxiliary equation method (SAEM246) and verified with the computer algorithm that developed for this study. Findings:For the NLSE, the distortion of the soliton solutions obtained by the SAEM246 method under different noise effects is clearly shown and simulated. Originality:This method has never been applied before to a NLSE containing a stochastic function to analyze the noise effect. The novelty of the problem and applied method has resulted in many new and original soliton solutions and their behaviors under the noise effects.

On the examination of optical soliton pulses of Manakov system with auxiliary equation technique

Purpose:This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. Methodology:After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear partial differential equation problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is formed and by aid of the any symbolic calculation package the unknown parameters are obtained. In order to gain the soliton solutions, the proposed series form, the solution functions of the method, and the cWT are brought together and it is checked that the obtained solutions satisfy the main equation. By considering the functions that satisfy the main equation as solutions, graphical presentations are made and interpreted in order to better understand the obtained results. Findings:By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright–dark have been obtained, and it has been shown that the method can be applied effectively for such equations. Originality:The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright soliton pulse solutions have not been reported before.

A COMPARATIVE ANALYSIS REPORT ON THE MULTI-WAVE FRACTIONAL HIROTA EQUATION IN NONLINEAR DISPERSIVE MEDIA

The propagation attributes of waves and its modeling maneuvers have a significant role in maritime, coastal engineering, and ocean. In the geographical fields, waves are primary source of environmental process owed to energy conveyance on the floating structure or on the synthetic field. This study aims to investigate the new auxiliary equation method to obtain analytical solutions of the nonlinear Hirota model with fractional order. The fractional model is developed by utilizing Riemann–Liouville, [Formula: see text], and the fractional-order Atangana–Baleanu differential operator in Riemann–Liouville sense. The solitonic patterns of the nonlinear fractional Hirota equation successfully surveyed, where the exact solutions are presented by rational, trigonometric, hyperbolic, and exponential functions. The contravene of surveyed results with the substantially recognized result is executed which states the novelty of obtained results. Three dimensional as well as two-dimensional comparison is presented for a couple of Hirota model solutions which are revealed diagrammatically for appropriate parameters by using Mathematica. We strongly believe that this study will help physicists to predict some new conceptions in the field of mathematical physics.

A Dispersive Formulation of Time-Domain Meshless Method With PML Absorbing Boundary Condition for Analysis of Left-Handed Materials

We have proposed a dispersive formulation of scalar-based meshless method for time-domain analysis of electromagnetic wave propagation through left-handed materials (LHMs). Moreover, we have incorporated Berenger’s perfectly matched layer (PML) absorbing boundary condition (ABC) into the dispersive formulation to truncate open-domain structures. In general, the dispersive formulations of conventional numerical methods are proper tools for analysis of left-handed (LH) media, whenever the LH media can be characterized by considering spatial or frequency dispersion effect for the constitutive parameters. In comparison to the grid-based numerical methods, it is proven that meshless methods not only are strong tools for accurate approximation of derivatives in Maxwell’s equations but also can provide more flexibility in modeling the spatial domain of problems. However, we have not seen any reports on using dispersive forms of meshless methods for simulation of wave propagation in metamaterials and applying any PML ABCs to dispersive formulation of meshless method. The proposed formulation enables us to take advantage of meshless methods in analysis of LH media with frequency-dependent parameters. For modeling the frequency behavior of the medium, we have used auxiliary differential equation (ADE) method based on the relations between electromagnetic fields intensities and current densities. Effectiveness of the proposed formulation is verified by analysis of 2-D and 3-D numerical examples; also, some basic factors which affect the accuracy, stability, and computational cost of the simulations are studied.

ANALYTICAL SOLUTIONS OF (1+1)- DIMENSIONAL DISTRIBUTED LONG WAVE (DLW) EQUATION WITH AUXILIARY EQUATION METHOD

In this article, the exact solutions of the (1+1)-dimensional (DLW) equation, a fractional partial differential equation in conformable sense, which is a nonlinear, are given. Furthermore, with the aid of the mathematica program it is seen that the analytical solutions revealed with the auxiliary equation method satisfies the equation.

New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

In this paper, we use general Riccati equation to construct new solitary wave solutions of the Van der Waals normal form, which is one of the most famous models for natural and industrial granular materials. It is very important to understand the static and dynamic characteristics of this models in many application fields. We solve the general Riccati equation through different function transformation, and many new hyperbolic function solutions are obtained. Then, it is substituted into the Van der Waals normal form as an auxiliary equation. Abundant types of solitary-wave solutions are obtained by choosing different coefficient in the general Riccati equation, and some of them have not been found in other documents. The results show that the analysis method we used is very simple and effective for dealing with nonlinear models.

Exact Solutions of the Nonlinear Modified Benjamin-Bona-Mahony Equation by an Analytical Method

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters.

Solitons’ behavior of waves by the effect of linearity and velocity of the results of a model in magnetized plasma field

The nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov model plays the behavior of weakly nonlinear ion-acoustic waves in a magnetized electron-positron plasma, containing the same capacity of cool and hot components of each species. In this article, we have investigated the mathematical model namely, (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation through the new auxiliary equation method for locating the closed-form travelling and solitary wave solutions. We have constructed a well-built 3D wave profile that signifies the variations in the same solution for particular values of the associated parameters. The novel point of this article is that, the generic values of the velocity parameter (λ) of the obtained solutions represent in the wave profile which thoroughly explains the reasons for the change in the variation of solitons. Besides, the type of wave profile subject to different values ​​of all other parameters that are related to the linearity of the stated model is also discussed. We have obtained numerous new wave solutions that establish the effectiveness of the implemented method in solving evolution equation which is nonlinear in nature and has higher dimension (HNEEs).These discussions may be useful or helpful in further research on this model. Besides, the implemented method is reliable for exploring important nonlinear waves that improve a variety of dynamic models arising in nonlinear science, mathematical physics, and engineering fields.

W-shaped profile and breather-like soliton of the fractional nonlinear Schrödinger equation describing the polarization mode in optical fibers

We use the fractional nonlinear Schrödinger equation (FNLSE) to describe the polarization mode in optical fiber with Self-Steepening, Self-Frequency Shift, and Cubic-quintic terms to analyze the effects of the fractional time parameter (FTP) on bright and dark solitons as well as breather-like solitons. We use the transformation hypothesis and auxiliary equations method to obtain three families of solutions such as combined bright soliton, dark solitons, and rational solitons. We have shown the effects of the fractional parameter (FP) on the W-shaped profile, bright and dark optical soliton solutions as well as the corresponding chirp component. It is observed that for small values of the FP, optical soliton shape is affected and the soliton is unstable. Moreover, one observes the effects of fraction time on Modulation Instability (MI) gain spectra and MI bands. For certain values of the FP, it is formed sides lobes and for specific small values of the FP, both stability zone increases and amplitude of the MI gain increase while the stability zones increase. To confirm the robustness of the analytical results, we have used a numerical investigation. One exhibits the formation of breathers-like soliton with stable amplitude for small values of the FTP. It results from this study that the FP is efficient and can be used as an energy source where soliton or breathers-like soliton are involved for communication in optical fibers.

Chiral solitons of (2+1)-dimensional stochastic chiral nonlinear Schrödinger equation

This paper studies [Formula: see text]-dimensional stochastic chiral nonlinear Schrödinger equation dynamically. A number of novel solutions such as periodic, singular, dark and bright solitons solutions are retrieved. The extraction of these solutions is helped by three efficient and robust integration tools such as the Kudryashov’s R function method, Jacobi elliptic function method (JEFM) and the modified auxiliary equations (MAE) method. The parametric constraints for the existence of such solutions are also listed. Moreover, 3D plots of few obtained solutions are presented in this paper.

Nonlinear self-adjointness, conserved quantities and Lie symmetry of dust size distribution on a shock wave in quantum dusty plasma

The current exploration explains new traveling wave profiles by the Lie symmetry approach Here, we derive the (3+1)-dimensional Zakharov–Kuznetsov burgers equation for the shock wave phenomena in plasma with dust-charged particles having quantum effects. The structures of this model are made by Abelian algebra. The considered equation is reduced to an ordinary differential equation by the use of this Abelian algebra. The complete derivation of the classical symmetries of the considered model and reduction corresponding to the first four symmetries has been added. The new auxiliary equation method is implemented for the formation of the new traveling wave structures. The graphical explanation of some obtained results are elaborated in detail. The nonlinear self-adjointness of the considered model is also part of this work. By this concept, we have evaluated the conservation laws of the model. This work is new and does not exist in literature.

Dispersive optical solitons of Biswas–Arshed equation with a couple of novel approaches

Purpose:The importance of efficient analytical methods in solving nonlinear partial differential equations and nonlinear optical differential equations cannot be underestimated. Because of this importance, to solve efficiently above mentioned partial differential equations, a method has been suggested to use the improved Kudryashov and a sub equation form of new extended auxiliary equation methods. In addition the modified form of method has also been given. With this choice, it has been aimed to benefit from the advantages of improved Kudryashov and sub equation of extended auxiliary equation methods at the same time without spending additional processing and time in problem solving, to obtain more solution functions and soliton types offered by the methods. Methodology:As a first step to implement the proposed method, the nonlinear ordinary differential form (NODE) of the nonlinear partial differential equation has been obtained, the algorithms of the approached methods have been presented and applied to the Biswas–Arshed problem, which is an important equation in optic. Then, the polynomial form of the problem has been obtained, the solution function of the examined problem has been formed by choosing the appropriate sets and their substitutions in the proposed solution functions. It has been checked that the solution functions satisfy the Biswas–Arshed equation and then the graphical representation of the obtained results have been given. Findings:The application of the methods has carried out successfully, and both the linear algebraic equations system, the possible solution sets and the solution functions of the main equation, have been obtained seamlessly. It has been observed that both methods can be applied successfully and effectively and more results can be obtained at the same time, without spending an additional procedure and time. It can explicitly seen from both the obtained solution functions and the graphical results that the advantages of both methods can be used at the same time and can be applied to many nonlinear problems. Originality:In summary, the originality of this work is that the Biswas–Arshed equation has not been solved with both methods separately before, and it has not been presented in a study showing that both methods can be used together.

A Study of the Soliton Solutions with an Intrinsic Fractional Discrete Nonlinear Electrical Transmission Line

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end.