Hyperbolic Solutions Research Articles

New Solutions to the Fractional Perturbed Chen Lee Liu Model with Time-Dependent Coefficients: Applications to Complex Phenomena in Optical Fibersns

In this paper, we consider the fractional perturbed Chen Lee Liu model with time-dependent coefficients (FPCLLM-TDCs). We apply the mapping method in order to get hyperbolic, elliptic, trigonometric and rational fractional solutions. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as fiber optics and plasma physics. Finally, we show how the conformable fractional derivative order affect the exact solutions of the FPCLLM-TDCs. Furthermore, we examine the effects of time-dependent coefficients when these coefficients take on special cases such as random variables, polynomials, and hyperbolic functions.

New dynamics performances to the established dark solitons in Polariton Condensate

Abstract New diverse enormous soliton solutions to the Gross–Pitaevskii Equation, which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping, have been constructed for the first time by using two various Schemes. The two utilized Schemes are the Generalized Kudryashov Scheme, the (G'/G)-Expansion Scheme. Throughout these two suggested schemata we construct new diverse forms solutions that include dark, bright-shaped soliton solutions, combined bright-shaped, dark-shaped soliton solutions, hyperbolic function soliton solution, singular shaped soliton solution and other rational soliton solutions. The two 2D and 3D figures designs have been configured by using the Mathematica-program. In addition, the Haar Wavelet Numerical Scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested Schemes to show existing similarity between the soliton solutions and numerical solutions.

Investigation of oceanic wave solutions to a modified (2+1)-dimensional coupled nonlinear Schrödinger system

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized [Formula: see text]-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

Propagation of wave insights to the Chiral Schrödinger equation along with bifurcation analysis and diverse optical soliton solutions

In this study, the modified Sardar sub-equation method is capitalised to secure soliton solutions to the (1+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional chiral nonlinear Schrödinger (NLS) equation. Chiral soliton propagation in nuclear physics is an extremely attractive field because of its wide applications in communications and ultrafast signal routing systems. Additionally, we perform bifurcation analysis to gain a deeper understanding of the dynamics of the chiral NLS equation. This highlights the complex behaviour of the system and exposes the conditions under which various types of bifurcations occur. Additionally, a sensitivity analysis is performed to assess how small changes in initial conditions and parameters influence the solutions, offering valuable perspectives on the stability and dependability of the acquired solutions. By employing the above-mentioned methodology, we derive a variety of exact solutions, including periodic, singular, dark, bright, mixed trigonometric, exponential, hyperbolic, and rational wave solutions. The study’s findings advance our theoretical knowledge of chiral NLS equations and have potential applications in optical communication and related fields.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. 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. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Investigation of Space-Time Dynamics of Akbota Equation using Sardar Sub-Equation and Khater Methods: Unveiling Bifurcation and Chaotic Structure

This paper focuses on obtaining exact solutions of nonlinear Akbota equation through the application of the modified Khater method and Sardar sub-equation method. Renowned as one of the latest and precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The equation is crucial in the study of optical solitons, which are stable pulses of light that maintain their shape over long distances. The Akbota equation helps in understanding the behavior and stability of these solitons. The governing equation undergoes transformation into an ordinary differential equation through a well-suited wave transformation. This analytical simplification paves the way for the derivation of trigonometric, hyperbolic, and rational solutions through the proposed methods. To illuminate the physical behavior of the model, the study presents graphical plots of the selected solutions of Khater and Sardar sub-equation method. This visual representation, achieved by selecting appropriate values for arbitrary parameters, enhances the understanding of the system’s dynamics. All calculations in this study are meticulously conducted using the Mathematica and Maple software, ensuring accuracy and reliability in the analysis of the obtained solution. Furthermore we investigate the sensitivity analysis of the dynamical system.

The impact of standard Wiener process on the optical solutions of the stochastic nonlinear Kodama equation using two different methods

The stochastic nonlinear Kodama (SNLK) equation forced in the Stratonovich sense by multiplicative noise is considered here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are acquired using ( G′/ G)-expansion method and mapping method. Because the SNLK equation is extensively used in is extensively used in nonlinear optics, fluid dynamics, and plasma physics, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3D and 2D graphs. We infer that multiplicative noise impacts and stabilizes the solutions of SNLK equation.

Novel solitary wave solutions for stochastic nonlinear reaction–diffusion equation with multiplicative noise

The primary objective of this paper is to explore solitary wave solutions within the context of the stochastic Reaction–Diffusion equation, a problem that has not been previously investigated with respect to multiplicative noise. This marks a novel aspect of the current study. To accomplish this goal, we utilize two distinct integration methods: the extended Kudryashov’s method and the G′G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{G'}{G}\\right) $$\\end{document}-expansion method. These methods are chosen for their efficacy in handling nonlinear differential equations and are applied in tandem to uncover new trigonometric and hyperbolic stochastic exact solutions. The employment of these methods is crucial in generating solutions that have not been previously identified in this particular context. Subsequently, to assess the validity and usefulness of these newly discovered solutions, we conduct numerical simulations. These simulations involve implementing the derived solutions across a variety of scenarios or samples, allowing for an evaluation of their performance and applicability. This paper represents a significant contribution to the field of stochastic Reaction–Diffusion equations by introducing a pioneering investigation into the effects of multiplicative noise on solitary wave solutions. Furthermore, it offers a fresh perspective on obtaining exact solutions in the presence of such noise, showcasing the potential of the integration methods in tackling complex nonlinear stochastic systems.

Investigating optical soliton pattern and dynamical analysis of Lonngren wave equation via phase portraits

This study focuses on finding effective solutions to a mathematical equation known as the Lonngren wave equation. The solutions from the Lonngren wave equation can be used to evaluate electromagnetic signals in cable lines and sound waves in stochastic systems. The main equation is transformed into an ordinary differential equation by utilizing a suitable wave transformation, allowing for the exploration of mathematical models by using the modified Khater technique to detect the exact solution of a solitary wave. We use the provided method to derive the trigonometric, rational, and hyperbolic solutions. To illustrate the model’s physical behavior, we also present graphical plots of selected solutions to illustrate the physical behavior of the model. By choosing appropriate values for arbitrary factors, the visual representation enhances the understanding of the dynamical system. Furthermore, the system is transformed into a planar dynamical system, and phase portrait analysis is conducted. Additionally, the sensitivity analysis of the dynamical system confirms that slight changes in the initial conditions will have minimal impact on the stability of the solution. The existence of chaotic dynamics in the Lonngren wave equation is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase portraits will be used to demonstrate these chaotic behaviors.

New exact solutions of the conformable space-time two-mode foam drainage equation by two effective methods

In this article, the two-mode foam drainage equation in terms of time and space conformable sense has been investigated. Two effective methods, the generalized exponential rational function method (GERFM) and the improved version of the Bernoulli sub-equation function method (IBSEFM), are used to get new solutions of underlying equation. The fractional travelling wave transformation is applied to convert nonlinear partial differential equations to nonlinear ordinary differential equations. Proposed methods successfully extract trigonometric, hyperbolic and exponential solutions. Some of the obtained solutions are visualized to understand the effect of fractional orders of time and space derivatives on the wave profile and the dynamic behavior of the solutions.

The expansion behavior of the hyperbolic solution in f(R,G) gravity

In this study, our primary focus lies in meticulously exploring the intricacies of the universe by employing a flat Friedmann–Lemaître–Robertson–Walker (FLRW) model within the framework of [Formula: see text] gravity. In our analysis, the [Formula: see text] function is delineated as the sum of two distinct components, commencing with a quadratic correction of the geometric term denoted by [Formula: see text], structured as [Formula: see text], alongside a matter term denoted by [Formula: see text], where [Formula: see text] and [Formula: see text] symbolize the Ricci scalar and Gauss–Bonnet invariant, respectively. In the pursuit of solutions to the gravitational field equations within the [Formula: see text] formalism, we embrace a specific expression for the scale factor, represented as [Formula: see text] [D. Rabha and R. R. Baruah, The dynamics of a hyperbolic solution in [Formula: see text] gravity, Astron. Comput. 45 (2023) 100761]. In this context, the parameters [Formula: see text] and [Formula: see text] intricately shape the scale factor’s behavior. The model posits the intriguing prospect of perpetual cosmic acceleration when [Formula: see text], signifying a continuous expansion of the universe. Conversely, for [Formula: see text], the model proposes a pivotal transition from an early deceleration phase to the present accelerated epoch, a transformative shift in line with our understanding of cosmic evolution. Furthermore, the model demonstrates its credibility by satisfying Jean’s instability condition during the shift from a radiation-dominated era to a matter-dominated era, substantiating the formation of cosmic structures. In our analysis, a central focus is directed toward scrutinizing the equation of state parameter [Formula: see text] within our model. We delve into a comprehensive examination of the scalar field and meticulously assess the energy conditions surrounding the derived solution. To establish the robustness of our model, we deploy an array of diagnostic tools, including the Jerk, Snap, and Lerk parameters, along with the Om diagnostic, Classical stability of the model, and statefinder diagnostic tools, Observational Constraints on the Model Parameters. The outcomes, intertwined with a detailed analysis of both the results and the inherent intricacies of the model, are diligently clarified and presented.

Dynamics of geometric shape solutions for space-time fractional modified equal width equation with beta derivative

The modified equal width equation describes the propagation of shallow water waves in which nonlinear and dispersive effects are significant, including phenomena such as wave breaking, soliton interactions, ion-acoustic waves, energy transfer in plasma, and nonlinear stress waves. The aim of this article is to establish some novel and generic solutions to the space-time fractional modified equal width (MEW) equation using the two variable (θ′/θ, 1/θ)-expansion technique, a modification of the (θ′/θ)-expansion method. A wide range of geometric shapes and inclusive soliton solutions have been constructed, comprising rational, trigonometric, and hyperbolic solutions, along with their integration to the equation under consideration. The two- and three-dimensional graphs of the solitons, including periodic, V-shaped, bell-shaped, singular periodic, flat kink-shaped, plane-shaped, peakon, and parabolic, illustrate the physical aspects of the solitary wave solution and the effect of the fractional parameter. The results demonstrate the efficiency, appropriateness, and reliability of the adopted technique for investigating fractional-order nonlinear evolution equations in science, technology, and engineering.

The star–square relation and the generalized star–triangle relation from 3d supersymmetric dualities I

We study duality transformations of the star–square relation and the generalized star–triangle relation for Ising-like lattice spin models. The lattice spin models are obtained via gauge/YBE correspondence which connects the supersymmetric gauge theories and lattice spin models of statistical mechanics. By the use of integral identities coming from the duality of three-dimensional supersymmetric gauge theories, we construct hyperbolic, lens hyperbolic, trigonometric, and rational solutions to the duality transformations. These duality transformations allow us to construct spin lattice models with four-spin (the star–square relation) and three-spin (the generalized star–triangle relation) interactions.

Travelling wave solutions and conservation laws of the (2+1)-dimensional new generalized Korteweg–de Vries equation

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F-expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W-shaped, M-shape, V-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

Dynamics of novel soliton and periodic solutions to the coupled fractional nonlinear model

This study secures the soliton solutions of the (2+1)-dimensional Davey–Stewartson equation (DSE) incorporating the properties of the truncated M-fractional derivative. The DSE and its coupling with other systems have extensive applications in many fields, including physics, applied mathematics, engineering, hydrodynamics, plasma physics, and nonlinear optics. Various solutions, such as dark, singular, bright-dark, bright, complex, and combined solitons, are derived. In addition, exponential, periodic, and hyperbolic solutions are also generated. The newly designed integration method, known as the modified Sardar subequation method (MSSEM), has been applied in this study for extracting the solutions. The approach is efficient in explaining fractional nonlinear partial differential equations (FNLPDEs) by confirming pre-existing solutions and producing new ones. Furthermore, we plot the density, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The outcomes of this work indicate the effectiveness of the method utilized to improve nonlinear dynamical behavior. We anticipate that our work will be helpful for a large number of engineering models and other related problems.

Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

Dynamical behavior of the fractional nonlinear Kadoma equation in plasma physics and optics

The nonlinear Kadoma equation with M-truncated derivatives (NLKE-MTD) is taken into consideration here. By using generalized Riccati equation method (GRE method) and Jacobi elliptic function method, new hyperbolic, rational, trigonometric and elliptic solutions are discovered. Because the NLKE is widely employed in optics, fluid dynamics and plasma physics, the resulting solutions may be used to analyze a wide variety of important physical phenomena. The dynamic behaviors of the different derived solutions are interpreted using 3D and 2D graphs to explain the effects of M-truncated derivatives. We may conclude that the surface moves to the right as the order of M-truncated derivatives increases.