Auxiliary Equation Method Research Articles

Optical solutions for perturbed conformable Fokas–Lenells equation via Kudryashov auxiliary equation method

This paper is dedicated to the study of optical soliton solutions for the perturbed Fokas–Lenells equation with conformable derivative using the Kudryashov auxiliary equation method. The studied optical solutions include a class of categories, comprising dark, mixed dark-bright, multi bell-shaped, bell-shaped, and wave optical solutions. Furthermore, we analyzed the magnitude of the perturbed conformable Fokas–Lenells equation by investigating the impact of the conformable parameter and the effect of the time parameter on the novel optical solutions. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results obtained illustrate that the proposed method is robust, efficient, and readily applicable for constructing new solutions to a wide range of nonlinear fractional partial differential equations. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics and mathematical physics.

Formation of solitary waves solutions and dynamic visualization of the nonlinear schrödinger equation with efficient techniques

This article investigates the non-linear generalized geophysical KdV equation, which describes shallow water waves in an ocean. The proposed generalized projective Riccati equation method and modified auxiliary equation method extract a more efficient and broad range of soliton solutions. These include U-shaped, W-shaped, singular, periodic, bright, dark, kink-type, breather soliton, multi-singular soliton, singular soliton with high amplitude, multiple periodic, multiple lump wave soliton, and flat kink-type soliton solutions. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Maple and Wolfram Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. Sensitivity and modulation instability analyses were performed to offer insights into the dynamics of the examined model. The observed dynamics of the proposed model were presented, revealing quasi-periodic chaotic, periodic systems, and quasi-periodic behaviour. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations.

Stability analysis and retrieval of new solitary waves of (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations using auxiliary equation technique

The Kadomtsev–Petviashvili (KP) equations are nonlinear partial differential equations which are widely used for the modeling of wave propagation in hydrodynamic and plasma systems. This study aims to make a valuable contribution to the literature by providing new solitary waves to the (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili (pKP)-B-type Kadomtsev–Petviashvili (BKP) equations. For this, the auxiliary equation method associated with Bernoulli equation is used and new solutions for the considered equations are obtained. The stability of obtained solutions is demonstrated using nonlinear analysis. It is shown that this method for the considered pKP–BKP equations is an important step forward in an overall mathematical framework for similar equations.

Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization.

By using the power-exponential function method and the extended hyperbolic auxiliary equation method, we present the explicit solutions of the subsidiary elliptic-like equation. With the aid of the subsidiary elliptic-like equation and a simple transformation, we obtain the exact solutions of Hirota equation which include bright solitary wave, dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions. Some of these solutions are found for the first time, which may be useful for depicting nonlinear physical phenomena. This approach can also be applied to solve the other nonlinear partial differential equations.

Investigation of soliton solutions for the NWHS model with temperature distribution in an infinitely long and thin rod

This study proposed a modified [Formula: see text]-expansion method to seek the new exact traveling wave solutions of the Newell–White–Head–Segel (NWHS) Model. This is an amplitude equation utilized for distributing temperature within a rod that is infinitely thin and long, or determining the flow velocity of a fluid through a pipe that is infinitely long but has a small diameter. The modified [Formula: see text]-expansion method is used to extract the new exact solutions. The solutions of this model are categorized in hyperbolic, trigonometric, and rational forms. Moreover, we compare our results with the new auxiliary equation method. The 3D, line and corresponding contour representation of these solutions are depicted by choosing the different values of parameters.

Optical soliton solutions of the perturbed fourth-order nonlinear Schrödinger-Hirota equation with parabolic law nonlinearity of self-phase modulation

This article introduces an examination of optical soliton solutions for the perturbed fourth-order nonlinear Schrödinger-Hirota equation, which plays a crucial role in optics. For the first time, it utilizes a novel approach by applying the extended auxiliary equation method. This equation models the propagation of optical pulses through nonlinear media, such as optical fibers, and has been the subject of many studies. Our goal extends beyond merely acquiring a significant number of soliton solutions using the method described in this article; we also aim to investigate the impact of the coefficients of group velocity dispersion, parabolic law, and fourth-order dispersion terms on soliton propagation in the problem examined. The 2D, 3D, and contour plots of the acquired dark and bright solitons, which represent the most fundamental soliton types, are presented. Additionally, all other calculations are performed using symbolic algebraic software. The results provide us with valuable insights, confirming that the introduced model can be analyzed from a physical perspective. It is demonstrated that the proposed technique is not only important but also efficient in analyzing various nonlinear scientific problems.

A computational scheme and its comparison with optical soliton solutions for the stochastic Chen–Lee–Liu equation with sensitivity analysis

In this study, the stochastic Chen–Lee–Liu equation is considered numerically and analytically which is forced by the multiplicative noise in the Itô sense. The Chen–Lee–Liu equation is a special type of Schrödinger’s equation which has applications in optical fibers and photonic crystal fibers. The stochastic Crank–Nicolson scheme is formed to obtain the computational results. The numerical scheme is analyzed under the mean square sense and Von-Neumann criteria to show consistence and stability, respectively. Meanwhile, stochastic optical soliton solutions are attained by using two techniques, namely, the modified auxiliary equation method and the generalized projective Riccati equation method. These methods provide us with the different types of optical soliton solutions such as hyperbolic, trigonometric, mixed trigonometric, and rational forms. Mainly, the comparison of computational results with newly constructed optical soliton solution is shown graphically. To compare these results, initial conditions and boundary conditions are constructed by selecting some soliton solutions. The 3D and line graphs are drawn by choosing different values of parameters. Additionally, the sensitivity analysis is observed for the different initial values.

Rotationally symmetric transverse magnetic vector wave propagation for nonlinear optics

In this paper the theory and simulation results are presented for 3D cylindrical rotationally symmetric spatial soliton propagation in a nonlinear medium using a modified finite-difference time-domain general vector auxiliary differential equation method for transverse magnetic polarization. The theory of 3D rotationally symmetric spatial solitons is discussed, and compared with two (1 + 1)D, termed “2D” for this paper, hyperbolic secant spatial solitons, with a phase difference of π (antiphase). The simulated behavior of the 3D rotationally symmetric soliton was compared with the interaction of the two antiphase 2D solitons for different source hyperbolic secant separation distances. Lastly, we offer some possible explanations for the simulated soliton behavior.

Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

Abstract The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

Hirota–Maccari system arises in single-mode fibers: abundant optical solutions via the modified auxiliary equation method

Hirota–Maccari system arises in single-mode fibers: abundant optical solutions via the modified auxiliary equation method

Exploring optical soliton solutions in highly dispersive couplers with parabolic law nonlinear refractive index using the extended auxiliary equation method

This study delves into the realm of optical soliton solutions within the intricate framework of highly dispersive couplers integrated with optical metamaterials featuring a parabolic law nonlinear refractive index. Employing the extended auxiliary equation method, the investigation systematically unveils various soliton solutions, including dark solitons, bright solitons, singular solitons, and a distinctive amalgamation of bright and singular solitons. The results illuminate the diverse dynamics of solitons in these complex systems, offering crucial insights into the interplay among dispersion, nonlinearity, and metamaterial characteristics. Beyond enhancing the theoretical understanding of soliton behavior in optical metamaterial couplers, the findings hold practical significance by potentially influencing the design and optimization of optical communication devices and systems.

The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis

In light of fractional theory, this paper presents several new effective solitonic formulations for the Langmuir and ion sound wave equations. Prior to this study, no previous research has presented the comparision and obtained the generalized fractional soliton solutions of this kind with power law kernel and Mittag-Leffler kernel. The ion sound and Langmuir wave equations are essential in plasma physics, offering insights into the collective behavior of charged particles in plasmas and enabling diagnostics and control of these complex, ionized gas systems. The two distinct fractional order differential operators are substituted for the traditional order derivative to reshape the examined model. The Atangana-Baleanu non-singular and non-local operator and conformable fractional operator are the fractional-order operators that are used to create the fractional complex system equations for Langmuir waves and ion sound. A constructive approach new auxiliary equation method utilizes to obtain the exact analytical soliton solutions for ion sound and Langmuir wave equation. A wide range of soliton solutions is obtained, including mixed complex solitary shock solutions, singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed singular solutions, exact solutions, mixed periodic solutions, and mixed hyperbolic solutions, dark soliton, bright soliton, trigonometric solutions, periodic results, and hyperbolic results. The solitons solution of the ion sound and Langmuir wave equations lies in their ability to maintain wave stability, their role in modeling wave propagation and nonlinear effects, their potential use as diagnostic tools, and their relevance in wave-particle interactions in plasma physics. The solitons provide a valuable framework for understanding the behavior of waves in plasmas and offer insights into the complex dynamics of these charged particle systems. A graphical comparison analysis of a few solutions is also shown here, taking into account appropriate parametric values through the use of the software package. Moreover, the results of this study have important implications for Hamilton's equations and generalized momentum, where solitons are employed in long-range interactions.

Explicit optical solitons of a perturbed Biswas–Milovic equation having parabolic-law nonlinearity and spatio-temporal dispersion

This paper deals with a new variant of the Biswas–Milovic equation, referred to as the perturbed Biswas–Milovic equation with parabolic-law nonlinearity in spatio-temporal dispersion. To our best knowledge, the considered equation which models the pulse propagation in optical fiber is studied for the first time, and the abundant optical solitons are successfully obtained utilizing the auxiliary equation method. Utilizing a wave transformation technique on the considered Biswas–Milovic equation, and by distinguishing its real and imaginary components, we have been able to restructure the considered equation into a set of nonlinear ordinary differential equations. The solutions for these ordinary differential equations, suggested by the auxiliary equation method, include certain undetermined parameters. These solutions are then incorporated into the nonlinear ordinary differential equation, leading to the formation of an algebraic equation system by collecting like terms of the unknown function and setting their coefficients to zero. The undetermined parameters, and consequently the solutions to the Biswas–Milovic equation, are derived by resolving this system. 3D, 2D, and contour graphs of the solution functions are plotted and interpreted to understand the physical behavior of the model. Furthermore, we also investigate the impact of the parameters such as the spatio-temporal dispersion and the parabolic nonlinearity on the behavior of the soliton. The new model and findings may contribute to the understanding and characterization of the nonlinear behavior of pulse propagation in optical fibers, which is crucial for the development of optical communication systems.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

Dynamical behavior of soliton solutions to the fractional phi-four model via two analytical techniques

This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein–Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it’s correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations.

Bright soliton of Stochastic perturbed Biswas-Milovic equation with cubic-quintic-septic law having multiplicative white noise

For the first time, the adopted stochastic form of the perturbed Biswas-Milovic equation with cubic-quintic-septic law having spatio-temporal and chromatic dispersion in the presence of multiplicative white noise in Ito sense was presented and examined. The Biswas-Milovic equation ˆ models numerous physical phenomena occurring in optical fiber. We analyzed the optical soliton solutions of the stochastic model with the aid of a subversion of the new extended auxiliary equation method. Furthermore, we investigated the evaluation of the noise impacts and the effects of some model parameters on the dynamics of the generated soliton. Finally, graphical depictions of the derived soliton types were represented for some solution functions. The stochastic model and the derived results will contribute to the comprehension of the nonlinear dynamics of pulse propagation in optical fibers which has great importance for the advancement of optical communication engineering.

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Dynamical study of groundwater systems using the new auxiliary equation method

In this research, the exact solitary wave solutions to the non-linear problem of underground water levels are found. This study examines the transport of solutes in groundwater systems with variable density flow. The mathematical equation that is used to explain how groundwater moves through an aquifer is known as the groundwater flow equation and is used in hydrogeology. The auxiliary equation method is used to gain the analytical solutions to the underlying model equation. These solutions are gained in the form of hyperbolic, trigonometric, exponential, and rational function solutions. Mathematica generates two-dimensional and three-dimensional graphs with suitable parameter values. The resulting solutions are also useful for researching wave interactions in several novel structures.

The auxiliary equation approach for solving reaction-diffusion equations

The paper concerns the auxiliary equation method and proposes an approach for finding the most general nonlinear term that can generalize a nonlinear differential equation, so that it keeps solutions expressed in terms of the same auxiliary equation. More precisely we will consider the second order reaction-diffusion equations and we will find the most general nolinear term of this type of equations, for which the solutions can be expressed in terms of the Riccati equation. The procedure is exemplified on Fitzhugh-Nagumo, Dodd-Bullough-Mikhailov, and Klein-Gordon models, seen as reaction-diffusion equations.

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.