Auxiliary Equation Research Articles

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.

The formation of invariant optical soliton structures to electric-signal in the telegraph lines on basis of the tunnel diode and chaos visualization, conserved quantities: Lie point symmetry approach

This study critically examines the nonlinear Lonngren wave model, a mathematical representation used to describe waves in shallow water environments such as beaches, lakes, and rivers, as well as electrical signals in semiconductor materials like tunnel diodes. The symmetry generators take into account the Lie invariance criteria. A three-dimensional Lie algebra can be obtained by using the suggested methodology. It links translation symmetries in space and time, respectively, with conservation of mass and energy. The third symmetry pertains to scaling or dilation. Recognizing the constraints of the inverse scattering transform in addressing the Cauchy problem for this equation, the investigation adopts new auxiliary equation approach to ordinary differential equations. This approach yields closed-form analytical solutions for the wave model. The obtained solutions include a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, smooth soliton, trigonometric solution, mixed singular solution, complex solitary and singular solution. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the total energy of the system in terms of its position and momentum. Chaos analysis is performed to describe periodic, quasi-periodic, and chaotic behaviours, and a sensitivity analysis highlights the model’s sensitivity to different wave numbers. Graphical representations of selected solutions are presented in both 3-D and 2-D, illustrating the behaviour under specific physical conditions. The study concludes by identifying a comprehensive set of local conservation laws for the nonlinear Lonngren wave equation, applicable to arbitrary constant coefficients, using conservation laws multipliers.

Solitary wave solutions for the (1+1)-dimensional nonlinear Kakutani–Matsuuchi model of internal gravity waves

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.

Nuclear softness in the variable moment of inertia model and its application to superdeformed bands in the mass region A ≈ 60 - 90

For superdeformed (SD) bands 58Ni (b1), 58Cu, 59Cu (b1), 61Zn, 62Zn, 65Zn, 68Zn, 84Zr, 86Zr (b1), 88Mo (b1, b2, b3) and 89Tc in the A ≈ 60 - 90 mass region, the nuclear softness (NS) parameter, σ, has been calculated using the VMINS3 model. The SD bands 58Ni (b1), 58Cu, 59Cu (b1), 62Zn, 65Zn, and 88Mo (b2, b3) have NS parameter values that are greater than those of the normal deformed bands, indicating smaller rigidity. The fluctuation of the NS parameter versus the gamma energy ratio, R, of SD bands in the A ≈ 60 - 90 mass region is one of the study's findings. The ratio of transition energies was used to calculate the band head spin, I0, by the Descartes method (the greatest technique to solve the quartic equation based on an auxiliary cubic equation) which was then confirmed by root mean square deviations. The estimated and observed transition energies are in good agreement.

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.

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

New traveling‐wave solutions for a nonlinear Dodd–Bullough–Mikhailov equation

In this article, we will present new and more general traveling‐wave solutions for the Dodd–Bullough–Mikhailov equation based on the extended Fan sub‐equation method. The two‐dimensional affine sphere's intrinsic geometry is determined by Dodd–Bullough in the three‐dimensional unimodular affine space. The most significant accomplishment of our method is that we are able to provide all forms of the previously obtained solutions in one action, using at least four different approaches. As a result, the general elliptic equation with five parameters can be connected to other sub‐equations with three parameters that already exist, such as the Riccati equation, the first sort of elliptic equation, the auxiliary ordinary equation, and the modified Riccati equation. The extended Fan sub‐equation method is utilized to construct traveling‐wave solutions, solitary wave solutions, dark soliton solutions, and bell soliton solutions for the nonlinear Dodd–Bullough–Mikhailov equation. Mathematica is utilized to depict the dynamics of various wave structures in 3D, 2D, and contour formats, utilizing a specific set of parameters. Our graphical comparative analysis suggests that the employed method is both reliable and powerful for obtaining exact solutions of nonlinear evolution equations. The approach employed in this study may also be applicable in enhancing the understanding of numerous other complex physical phenomena.

Exclusive soliton solutions arise in mono-mode optical fibre connecting to nonlinear Fokas system

Nonlinear partial evolution equations are mostly significant to illustrate critical phenomena in wave theory concerning real-world problems. The current study deals with the (2 + 1)-dimensional nonlinear Fokas model depicting the nonlinear pulse through the mono-mode optical fibers. Improved auxiliary equation and improved tanh schemes are executed on the considering governing system. Subsequently, a variety of optical soliton solutions with the nature of dynamic nonlinear waves are made accessible throughout the present exploration. Some of constructed solutions are figured out in 3D, 2D and contour sense for the visualization to the readers for making them understand of the characteristics of dynamic waves. The solitons are visible to be bright, dark, kink, anti-kink, singular kink, periodic, compacton, anti-compacton etc. in the current exploration. Involved free parameters are assigned with various numerical values and brought out the effects of nonlinear pulses in wave propagation along mono-mode optical fibers. The entire work might claim to be recorded in the literature as new aspects of research.

Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations

This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case.

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.

Novel solitonic structure, Hamiltonian dynamics and lie symmetry algebra of biofilm

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

Flow2Mesh: A flow-guided data-driven mesh adaptation framework

Mesh adaptation is crucial in numerical simulation, providing optimal resource allocation for accurately capturing physical phenomena. However, when applied to Computational Fluid Dynamics (CFD) problems with complex multi-scale properties, existing adaptation methods face huge challenges due to the high computational cost of solving auxiliary partial differential equations (PDEs) and the difficulty in aligning the flow features with mesh geometric features. In this work, an end-to-end data-driven mesh adaptation framework, Flow2Mesh, is proposed to address these challenges by adopting a hybrid modeling strategy to construct the mapping from pixelated flow-fields to graph-based meshes. It achieves a rapid and accurate one-step mesh adaptation via a perceptual feature network (PFN) and a mesh movement network (MMN). PFN extracts the global perceptual features from flow-fields to enhance flow feature representation and mesh resolution independence. In MMN, these features are utilized to deform the initial mesh to a topology-invariant adaptive mesh by a proposed physically driven mesh convolutional network. It considers the inherent mesh geometric information for efficient node feature aggregation and alignment of mesh density with a flow-field structure. To generate high-quality adaptive meshes, various mesh-related losses are designed to regularize the mesh movement and alleviate the mesh tangling. Experiments in CFD scenarios demonstrate the generalization of our model to different design parameters and mesh configurations. It takes three orders of magnitude less time to generate similar meshes than the PDE-based method. The results exhibit the potential of Flow2Mesh to be a flexible and reliable tool for rapid mesh adaptation in scientific and industrial fields.

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.

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.

Bifurcation analysis and exact wave solutions of the nano-ionic currents equation: Via two analytical techniques

In this work, numerous nonlinear evolution equations, namely the nano-ionic current equation, have wide applications in various fields of science and technology. To achieve this, we employ the generalized auxiliary equation and extended hyperbolic function techniques to investigate the exact wave solutions of the model under consideration. We obtained a range of exact wave solutions, including bright, dark, singular-periodic, periodic, and w-shape solutions, for the nonlinear model under examination, as well as rational and exponential function solutions with various structures, which are explored in our manuscript. In this paper, we compared the solutions obtained using the GAE and EHF approaches and also compared the obtained solutions using the GAE scheme with the solutions reported in Aljahdaly et al. [43]. Moreover, the three-dimensional and contour graphs are included to illustrate the structure of some obtained solutions by selecting specific parameter values. Bifurcation analysis is illustrated by phase plane analysis; phase plots are employed to depict the dynamic structure and are also analyzed for the existence of other solutions. These techniques serve as reliable, simple, and potent tools to analyze various nonlinear evolution equations found in physics, applied mathematics, and engineering.

An extended isogeometric boundary element formulation for three-dimensional linear elastic fracture mechanics

This paper presents a novel extended isogeometric boundary element formulation (XIGABEM) for three-dimensional linear elastic fracture mechanics. The formulation utilises the Dual BEM to accommodate coincident geometries for opposing crack surfaces, and inherits the well-known advantages of the NURBS basis as other isogeometric implementations. The originality herein involves the extension of the above-mentioned scheme to 3D using enrichment functions derived from asymptotic solutions for near-field crack tip displacements, in which Williams’ expansions are used on the crack surfaces and on the boundaries crossed by the crack front. Besides, Heaviside functions enrich external boundaries and allow the displacement discontinuity modelling. As with most enriched formulations, additional degrees of freedom are introduced; novel strategies are presented for the generation of auxiliary equations to recover a square system.Another key element of the proposed scheme is that the stress intensity factors are recovered directly from the solution vector and no post processing is required. Four applications demonstrate the formulation robustness, with results of models having comparatively few degrees of freedom comparing well against classical and other published results.

Lie-group modeling and simulation of a spherical robot, actuated by a yoke–pendulum system, rolling over a flat surface without slipping

The present paper aims at introducing a mathematical model of a spherical robot expressed in the language of Lie-group theory. Since the main component of motion is rotational, the space SO(3)3 of three-dimensional rotations plays a prominent role in its formulation. Because of friction to the ground, rotation of the external shell results in translational motion. Rolling without slipping implies a constraint on the tangential velocity of the robot at the contact point to the ground which makes it a non-holonomic dynamical system. The mathematical model is obtained upon writing a Lagrangian function that describes the mechanical system and by the Hamilton minimal-action principle modified through d’Alembert virtual work principle to account for non-conservative control actions as well as frictional reactions. The result of the modeling appears as a series of non-holonomic Euler–Poincaré equations of dynamics plus a series of auxiliary equations of reconstruction and advection type. A short discussion on the numerical simulation of such mathematical model complements the main analytic-mechanic development.