Auxiliary Equation Mapping Research Articles

Inquisition of optical soliton structure and qualitative analysis for the complex-coupled Kuralay system

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined.

Exact wave solutions of new generalized Bogoyavlensky–Konopelchenko model in fluid mechanics

In this paper, we examine a novel generalized [Formula: see text]-dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, [Formula: see text]-expansion method, [Formula: see text]-expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves.

Optical soliton solutions of fokas system and (2 + 1) Davey-Stewartson system by mapping method

In this article, a very effective technique called generalized auxiliary equation mapping method has been employed to investigate some very important nonlinear equations in optical fibers such as Fokas system and (2+1) Davey-Stewartson (DS) system. Under different situations, the obtained solutions exhibit various wave pattern like bright and dark soliton, kink soliton, periodic wave soliton and singular solitons. These solutions are novel and interesting and prove the efficiency of the method. The accuracy of the obtained results provides the competence of the method and ensures that it can be used for other mathematical models involved in optical fibers. Graphical simulation of some reported results has been discussed here to visualize and support the mathematical results in terms of 3D, 2D and contour plots.

The dynamical structures of the Sharma–Tasso–Olver model in doubly dispersive medium

In this article, the Sharma–Tasso–Olver model is investigated which characterizes the nonlinear double-dispersive evolution dispersive waves’ dynamical propagation in heterogeneous mediums. To study the dynamical nature of moving waves with a periodic and isolated nature analytically, the governing model is converted into an ordinary differential equation via a wave transformation to employ the extended modified auxiliary equation mapping approach and the G′G2 expansion method. In order to validate the computations, the stability of the obtained results is also established. It has been found that the model supports nonlinear solitary waves, periodic waves, shock waves and stable oscillatory waves. For a set of appropriate parameters, Mathematica is used to represent the dynamics of various wave structures as 3D, 2D and contour visualizations. Additionally, we may remark that the results discussed here are novel and innovative.

Some new wave solutions and modulation instability of a Hamiltonian amplitude equation in optical fibres

The nonlinear Schrödinger equation (NLSE) is one of the most important physical models for explaining the dynamics of optical soliton proliferation in optical fibre theory. Due to an extensive variety of applications for ultrafast signal routing systems and short light pulses in communications, optical soliton propagation in nonlinear fibres is a topic of substantial present interest. To overcome the ill-posedness of the unstable NLSE, a new term Lxt is introduced which leads to the Hamiltonian amplitude equation. In this work, we study exact wave solutions to the nonlinear complex model by utilizing a set of analytical approaches namely as the exp(−ξ(δ)) method, the improved F-expansion technique, and the extended modified auxiliary equation mapping method. Numerous solutions, including singular periodic, multi-periodic, single bell-shaped, and multi-bell-shaped wave frameworks were found. The proposed plan of action is swift, strong, as well as offers the requirements needed to verify the origin of these solutions. Additionally, a dynamics representation of the fascinating behaviour of several solidarity using 2D, 3D, and contour graphs is provided, also we discuss the stability analysis and modulation instability of the proposed model.

On the solitonic wave structures and stability analysis of the stochastic nonlinear Schrödinger equation with the impact of multiplicative noise

The nonlinear Schrödinger equation (NLSE) is a fundamental and significant physical model that plays a crucial role in demonstrating the dynamics of optical solitons in the theory optical fibers. The propagation of solitons in nonlinear optical fibers has garnered substantial interest due to its broad spectrum of applications, particularly in ultrafast signal routing systems and short light pulses for communications. This article focuses on investigating the impact of multiplicative noise on the solutions of this governing model. To achieve this, three distinct methods, namely the enhanced modified extended tanh expansion (METE) method, the generalized Kudryashov (GK) approach and the extended modified auxiliary equation mapping (EMAEM) method are employed to obtain a new variety of stochastic solutions. The perception of how noise intensity impacts the transmission of waves is a fundamental and critical issue in various fields. In the context of this study, an equation describes an exact wave pattern under the condition that s (representing noise intensity) equals zero. In such cases, the observed pattern remains undisturbed and matches the expected theoretical wave pattern. However, as the strength of noise s increases, the observed pattern begins to deviate from the ideal wave pattern and exhibits signs of damage or distortion. Furthermore, the stability analysis for the stochastic NLSE is performed, providing valuable insights into the behavior solutions as the strength of multiplicative noise increases. The validation of computational results through stability analysis enhances the confidence in the findings of the study. All the mathematical calculations, as well as graphical representations, have been meticulously achieved to present a comprehensive and detailed understanding of the impact of multiplicative on soliton propagation in nonlinear optical fibers.

Optical Dromions for Spatiotemporal Fractional Nonlinear System in Quantum Mechanics

In physics, mathematics, and other disciplines, new integrable equations have been found using the P-test. Novel insights and discoveries in several domains have resulted from this. Whether a solution is oscillatory, decaying, or expanding exponentially can be observed by using the AEM approach. In this work, we examined the integrability of the triple nonlinear fractional Schrödinger equation (TNFSE) via the Painlevé test (P-test) and a number of optical solitary wave solutions such as bright dromions (solitons), hyperbolic, singular, periodic, domain wall, doubly periodic, trigonometric, dark singular, plane-wave solution, combined optical solitons, rational solutions, etc., via the auxiliary equation mapping (AEM) technique. In mathematical physics and in engineering sciences, this equation plays a very important role. Moreover, the graphical representation (3D, 2D, and contour) of the obtained optical solitary-wave solutions will facilitate the understanding of the physical phenomenon of this system. The computational work and conclusions indicate that the suggested approaches are efficient and productive.

Optical soliton solutions, bifurcation, and stability analysis of the Chen-Lee-Liu model

The Chen-Lee -Liu model has many applications in assorted fields, particularly in the study of nonlinear dynamics, chaos theory, circuit design, signal processing, secure communications, encryption and decryption of chaotic signals, as well as cryptography. The modified extended auxiliary equation mapping method has been applied to the Chen-Lee-Liu model in this article and explores new wave profiles, such as singular periodic solutions, periodic solutions, and kink-type soliton solutions. The complex wave conversion is considered to make a simple differential equation. Three- and two-dimensional images are plotted using Mathematica and MATLAB, and their dispersion and nonlinearity effects are discussed. We also discuss the bifurcation analysis of the studied model. The stability of the equilibrium points is studied, and the phase portrait of the system is presented graphically. The obtained wave profiles might play an important role in telecommunication systems, fiber optics, and nonlinear optics.

On the dynamics of the (2+1)-dimensional chiral nonlinear Schrödinger model in physics

The nonlinear Schrödinger equation is one of the significant nonlinear complex models describing the optical solitons in dispersive media. In this study, the (2+1)-dimensional chiral nonlinear Schrödinger model has been investigated analytically which is of key importantance in the field of fluid sciences. A variety of exclusive travelling waveform solutions have been established for the complex dynamical model by employing a set of eminent analytical approaches namely the extended modified auxiliary equation mapping method, the improved F-expansion method, and the unified method, respectively. We obtained different forms of solitary types solutions with the help of these technique. The outcomes are a set of bell-shaped, single periodic, optical, and multi-periodic solutions. Finally, the stability of the developed results is also established to validate the computations. The study provides a very spectacular and appropriate way to put together several interesting wave demonstrations for more complex models of current era.

On some soliton structures to the Schamel–Korteweg-de Vries model via two analytical approaches

The Schamel–Korteweg-de Vries (S-KdV) model is used to predict the influence of surface for deep water in the presence of solitary waves. The aim of the study is to study the governing model analytically by employing the extended modified auxiliary equation mapping approach and the extended FAN sub-equation method. The 3D, 2D and contour plots are drawn to demonstrate the physical nature of the nonlinear model for a set of parameters. As a result, dark solitons, light solitons, solitary waves, periodic solitary waves, rational functions, and elliptic function solutions are established. Furthermore, the the developed results are verified with the aid of latest computing tool such as Mathematica or Maple. The applied strategy appears to be a more powerful and efficient scheme for achieving exact solutions to a number of diversified contemporary models of recent eras.

Optical and analytical soliton solutions to higher order non-Kerr nonlinear Schrödinger dynamical model

This article studies the propagation of the short pulse optical model governed by higher order nonlinear Schrödinger equation (HNLSE) with non-Kerr nonlinearity. The model is used to describe the propagation of ultrashort photons in highly nonlinear media. Upon establishing the general framework, we discuss the statics and dynamics of HNLSE by employing an extended modified auxiliary equation mapping (EMAEM) architectonic to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Obtained optical soliton solutions are analysed graphically to represent features like as width, amplitude, and structure of solitons. We will also discuss our governing model for M-shaped, Homoclinic breathers, multiwave, kink cross rational, and interaction of M-shaped with 1 and 2 kink solutions with the help of various ansatz transformations.

Dispersive optical solitons along with integrability test and one soliton transformation for saturable cubic-quintic nonlinear media with nonlinear dispersion

Three well known properties of solitons will be discussed in this paper, like Painleve test (P-test), Hirota bilinear method (HBM) and extended modified auxiliary equation mapping (EMAEM) architectonic in saturable cubic quintic nonlinear media (SCQNM) in the presence of nonlinear dispersion. The P-test will be used to examine the integrability of our governing model. For finding multiple soliton interaction HBM scheme will be adopted. EMAEM architectonic will be implemented in order to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Graphical outcomes (3D, 2D, and density plots) will be shown to reflect the actual behavior of results.

Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

Construction of abundant novel analytical solutions of the space–time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods

Abstract The space–time fractional generalized equal width (GEW) equation is an imperative model which is utilized to represent the nonlinear dispersive waves, namely, waves flowing in the shallow water strait, one-dimensional wave origination escalating in the nonlinear dispersive medium approximation, gelid plasma, hydro magnetic waves, electro magnetic interaction, etc. In this manuscript, we probe advanced and broad-spectrum wave solutions of the formerly betokened model with the Riemann–Liouville fractional derivative via the prosperously implementation of two mathematical methods: modified elongated auxiliary equation mapping and amended simple equation methods. The nonlinear fractional differential equation (NLFDE) is renovated into ordinary differential equation by the composite function derivative and the chain rule putting together along with the wave transformations. We acquire several types of exact soliton solutions by setting specific values of the personified parameters. The proposed schemes are expedient, influential, and computationally viable to scrutinize notches of NLFDEs.

Some new dispersive dromions and integrability analysis for the Davey–Stewartson (DS-II) model in fluid dynamics

This paper focuses on the Davey–Stewartson (DS)-II equation, and the extended modified auxiliary equation mapping (EMAEM) architectonic is used to develop a new set of solutions such as kink, singular kink, rational, combined soliton-like solutions, bell-type solutions, trigonometric and hyperbolic solutions. Furthermore, this study reveals that the used technique is efficient for solving other nonlinear evolution equations (NLEEs). The Painleve test ([Formula: see text]-test) will also be used to examine the integrability of the DS-II equation. Finally, graphical simulations are designed to show the exact behavior of solutions as well as the efficacy of the suggested strategy.

On the solitary wave solution of the viscosity capillarity van der Waals p-system along with Painleve analysis

This work proposes two well known schemes, painleve analysis test and auxiliary equation mapping (AEM) algorithm to find new solitary wave solutions for van der Waals p-system (VDW) in the viscosity capillarity version. For the clarity of physical features, new solitary wave solutions like periodic, kink and anti kink type, doubly periodic, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions, etc. are articulated coupled with graphically patterns 2D, 3D and density plots to visualize the dynamics of our model. This study reveals that the methods used are simple, reliable, unambiguous and concise as compared to many other techniques for solving nonlinear partial differential equations (NLPDE). A new direct way to solve NLPDEs is provided using this approach. The method allows one to obtain new accurate solutions of the solitary wave that cannot be obtained using other methodologies, and it can be implemented on a computer using computer algebraic systems such as Mathematica or Maple.

On global behavior for complex soliton solutions of the perturbed nonlinear Schrödinger equation in nonlinear optical fibers

In this research article, the perturbed nonlinear Schrödinger equation (P-NLSE) is examined by utilizing two analytical methods, namely the extended modified auxiliary equation mapping and the generalized Riccati equation mapping methods. Consequently, we establish several sorts of new families of complex soliton wave solutions such as hyperbolic functions, trigonometric functions, dark and bright solitons, periodic solitons, singular solitons, and kink-type solitons wave solutions of the P-NLSE. Using the mentioned methods, the results are displayed in 3D and 2D contours for specific values of the open parameters. The obtained findings demonstrate that the implemented techniques are capable of identifying the exact solutions of the other complex nonlinear evolution equations (C-NLEEs) that arise in a range of applied disciplines.

Dromions for the coupled Maccari’s system in fluid mechanics

• Coupled Maccaris (CM’s) system is discussed. • Generalized Auxiliary Equation Mapping algorithm is employed. • Painleve test ( P -test) is used to check integrability of model. • Some new dormions is achieved successfully. Present communication focuses on the coupled Maccari’s (CM’s) system, a Generalized Auxiliary Equation Mapping (GAEM) algorithm is implemented to get a new set of solutions like periodic and doubly periodic, dark, bright, trigonometric, singular, rational, combined soliton like solutions, hyperbolic solutions, etc. Furthermore, it is analyzed that the suggested scheme is efficient and effective for solving nonlinear evolution equations (NLEEs). The integrability of CM’s through Painleve test (P-test) is also addressed. Finally, graphical simulations (2D and 3D) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

Exact and solitary wave solutions of conformable time fractional Clannish Random Walker’s Parabolic and Ablowitz-Kaup-Newell-Segur equations via modified mathematical methods

The nonlinear fractional differential equations (FDEs) composed by mathematical modeling through nonlinear corporeal structures. The study of these kind models has an energetic position in different field of applied sciences. The dominant intension of this article is to extract novel analytical wave solutions of the conformable time fractional Clannish Random Walker’s Parabolic (CRWP) and (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equations in the sense of conformable derivative with the aid of modified mathematical methods, called modified extended auxiliary equation mapping and modified F-expansion schemes. By the virtue of employed techniques, different types of solutions are obtained in the form of trigonometric, hyperbolic, exponential and rational functions respectively. To prompt the essential propagated features, some investigated solutions are exhibited in form of 3D and 2D is planned by passing on the precise values to the parameters under the constrain conditions. The accomplished solutions show that these presented schemes are reliable, applicable and efficient.

Multiple optical soliton solutions for CGL equation with Kerr law nonlinearity via extended modified auxiliary equation mapping method

In this article, multiple optical soliton solutions are obtained using more intrinsic and suitable conditions on complex Ginzburg-Landau equation with Kerr law nonlinearity. The extended modified auxiliary equation mapping method is applied on this model in this regard. Optical soliton solutions such as, kink wave, dark soliton, bright soliton, singular soliton and bright and dark soliton solutions are explored with the help of three parameters which are of key importance for this strategy. In order to show the wave pattern of obtained solutions, three dimensional plots and corresponding two dimensional contours are also presented. Moreover, the prescribed strategy is more powerful and proficient to find out multiple soliton solutions for various nonlinear partial differential equations.