3D Graphical Representation Research Articles

Bifurcation analysis, phase portraits and optical soliton solutions of the perturbed temporal evolution equation in optical fibers

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

Dy-SheHeRASADe: A representation of the [formula omitted] sheet dynamics through surface descriptors

Molecular dynamics (MD) simulations are important tools for studying the dynamic motions of macromolecules at the atomic level. With the increasing capabilities of high performance computing, MD simulations are becoming more widely used. This allows molecular modelers to simulate the molecular behavior of large molecular architectures for much longer trajectories. Appropriate visualization of MD trajectories is becoming essential to provide an immediate and intuitive understanding of a molecule’s dynamics and function. In this study, we implement a novel 3D graphical representation, Dynamical Sheets Helper for RepresentAtion of SurfAce Descriptors (Dy-SheHeRASADe), to visualize the β sheet secondary structures of proteins in the context of molecular dynamics. Dy-SheHeRASADe is developed in UnityMol, an open source molecular viewer and prototyping platform. We considered β sheet fluctuations and hydrogen bond formation during molecular dynamics simulations to characterize the parts of β sheets with large motions or with labile bonds. We propose two visualization modes based on a surface representation of the β sheets calculated according to the positions of the α carbons and the hydrogen bonds between the β strands. The volumetric mode, in which this surface is enclosed in a semi-transparent volume that represents the fluctuation zone of the sheet during dynamics. The heatmap mode, in which the surface is colored according to the amplitude values of the α carbons. In addition, we quantify the β sheet fluctuations by displaying the values of the largest and smallest movements of the β sheets, the surface area of the sheets, and the number of hydrogen bonds.

Trend Stability Assessment for Hydrological Drought in Euphrates Basin (Türkiye) Using Triple Wilcoxon Test and Innovative Trend Analysis Methods

This study investigates the stability of hydrological drought trends in the Euphrates Basin from 1960 to 2020 using three-dimensional (3D) graphical representations based on innovative trend analysis (ITA) and triple Wilcoxon test (WT) methods. Unlike traditional ITA and WT, which are widely used for trend identification but do not inherently provide trend stability information, this study employs a novel approach to assess and visualize trend stability. The Triple WT method divides the data into three equal segments, examining differences without altering the time series. Drought indices are calculated for 3-month, 6-month, and 12-month time scales using historical streamflow data from five stations. The research identifies trends and their stabilities across three distinct periods: 1967–1984, 1985–2002, and 2003–2020. Results show that as the time scale increases, trend differences between extreme drought conditions diminish. One station consistently exhibits significantly decreasing trends, while three stations show unstable trends with notable variations in the standardized streamflow index (SSFI). The use of 3D-ITA and Triple WT effectively captures the dynamics and stability of drought trends, offering a deeper understanding of hydrological drought in the Euphrates Basin. These findings provide a reference for future studies on drought trend mechanisms in various climatic regions.

Unveiling the Dynamics of Nonlinear Landau-Ginzburg-Higgs (LGH) Equation: Wave Structures through Multiple Auxiliary Equation Methods

This comprehensive investigation delves deeply into the intricate dynamics governed by the nonlinear Landau-Ginzburg-Higgs equation. It uncovers a diversity of semi-analytical solutions by leveraging three auxiliary equation methods within the traveling wave framework. This article effectively utilizes the improved Kudryashov, Kudryashov's R, and Sardar's subequation methods. The methods discussed are advantageous because they are easy to implement and suitable for use with the Mathematica package program. Each method yields a distinct set of solutions, scrutinized across all cases. We elucidate the complex wave structures through 3D, 2D, and contour graphical representations, providing profound insights into their underlying characteristics. Furthermore, we scrutinize the influence of parameter variations on these wave structures, thereby offering a comprehensive understanding of their dynamic behavior.

An explainable model for predicting Worsening Heart Failure based on genetic programming

Heart Failure (HF) poses a challenge for our health systems, and early detection of Worsening HF (WHF), defined as a deterioration in symptoms and clinical and instrumental signs of HF, is vital to improving prognosis. Predicting WHF in a phase that is currently undiagnosable by physicians would enable prompt treatment of such events in patients at a higher risk of WHF. Although the role of Artificial Intelligence in cardiovascular diseases is becoming part of clinical practice, especially for diagnostic and prognostic purposes, its usage is often considered not completely reliable due to the incapacity of these models to provide a valid explanation about their output results. Physicians are often reluctant to make decisions based on unjustified results and see these models as black boxes. This study aims to develop a novel diagnostic model capable of predicting WHF while also providing an easy interpretation of the outcomes. We propose a threshold-based binary classifier built on a mathematical model derived from the Genetic Programming approach. This model clearly indicates that WHF is closely linked to creatinine, sPAP, and CAD, even though the relationship of these variables and WHF is almost complex. However, the proposed mathematical model allows for providing a 3D graphical representation, which medical staff can use to better understand the clinical situation of patients. Experiments conducted using retrospectively collected data from 519 patients treated at the HF Clinic of the University Hospital of Salerno have demonstrated the effectiveness of our model, surpassing the most commonly used machine learning algorithms. Indeed, the proposed GP-based classifier achieved a 96% average score for all considered evaluation metrics and fully supported the controls of medical staff. Our solution has the potential to impact clinical practice for HF by identifying patients at high risk of WHF and facilitating more rapid diagnosis, targeted treatment, and a reduction in hospitalizations.

Optical soliton solutions of the nonlinear complex Ginzburg-Landau equation with the generalized quadratic-cubic law nonlinearity having the chromatic dispersion

In this study, we consider the complex Ginzburg-Landau equation with the generalized quadratic-cubic law of self-phase modulation. This model finds applications in various fields, such as the study of superconductivity, nonlinear optical phenomena, pattern formation, and designing photonic devices and systems. This manuscript successfully employs the new Kudryashov method to derive analytical solutions for complex Ginzburg-Landau equations with the generalized quadratic-cubic law of self-phase modulation. The 3D, contour, and 2D graphical representations of the acquired solutions are represented. Therefore, W-shaped, bright, and dark soliton structures are derived. Through rigorous analysis and interpretation, valuable insights into the influence of the parameters of the presented model on the soliton behavior are achieved.

Pioneering the plethora of soliton for the (3+1)-dimensional fractional heisenberg ferromagnetic spin chain equation

Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

On multiparametrized integral inequalities via generalized α‐convexity on fractal set

This article explores integral inequalities within the framework of local fractional calculus, focusing on the class of generalized ‐convex functions. It introduces a novel extension of the Hermite‐Hadamard inequality and derives numerous fractal inequalities through a novel multiparameterized identity. The primary aim is to generalize existing inequalities, highlighting that previously established results can be obtained by setting specific parameters within the main inequalities. The validity of the derived results is demonstrated through an illustrative example, accompanied by 2D and 3D graphical representations. Lastly, the paper discusses potential practical applications of these findings.

Abundant Soliton Solutions to the Generalized Reaction Duffing Model and Their Applications

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effective techniques, the mapping method and Bernoulli sub-ODE technique, are used for the first time to obtain the soliton solutions of the proposed model. Initially, the traveling wave transform, which comes from Lie symmetry infinitesimals, is applied, and a nonlinear ordinary differential equation form is derived. These approaches effectively retrieve a hyperbolic, Jacobi function as well as trigonometric solutions while the appropriate conditions are applied to the parameters. Numerous innovative solutions, including the kink wave, anti-kink wave, bell shape, anti-bell shape, W-shape, bright, dark and singular shape soliton solutions, were produced via the mapping and Bernoulli sub-ODE approaches. The research includes comprehensive 2D and 3D graphical representations of the solutions, facilitating a better understanding of their physical attributes and proving the effectiveness of the proposed methods in solving complex nonlinear equations. It is important to note that the proposed methods are competent, credible and interesting analytical tools for solving nonlinear partial differential equations.

WebCM: A Web-Based Platform for Multiuser Individual-Based Modeling of Multicellular Microbial Populations and Communities.

WebCM is a web platform that enables users to create, edit, run, and view individual-based simulations of multicellular microbial populations and communities on a remote compute server. WebCM builds upon the simulation software CellModeller in the back end and provides users with a web-browser-based modeling interface including model editing, execution, and playback. Multiple users can run and manage multiple simulations simultaneously, sharing the host hardware. Since it is based on CellModeller, it can utilize both GPU and CPU parallelization. The user interface provides real-time interactive 3D graphical representations for inspection of simulations at all time points, and the results can be downloaded for detailed offline analysis. It can be run on cloud computing services or on a local server, allowing collaboration within and between laboratories.

Novel dynamics of the Fokas-Lenells model in Birefringent fibers applying different integration algorithms

Abstract The Fokas-Lenells model has the broad applications in nonlinear physics to study various soliton phenomena. Employing the direct algebraic scheme, the modified rational sine-cosine technique, and the (1/G′) expansion scheme, the analytical solutions to this model are derived. Double periodic waves, bright soliton, dark soliton, single and multiple breather waves, and periodic breather waves are extracted from this model using symbolic computation. The dynamic behaviors of the acquired outcomes are vividly illustrated through density, two-dimensional (2D), and three-dimensional (3D) graphical representations. These discoveries are strategically positioned to significantly contribute to the advancement of exploring nonlinear models, standing as a fundamental pillar for forthcoming research endeavors.

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

NEW WAVE SOLUTIONS OF TIME FRACTIONAL CHAFEE-INFANTE EQUATION WITH BETA DERIVATIVE

In this article, we discuss the exact solutions forthe Chafee-Infante equation involving beta fractional derivative. Beta fractional derivative which is a local derivative, is a modification of conformable fractional derivative. Using the Modified Kudryashov Method, we obtain the general solution of the time fractional Chafee-Infante equation with the help of Wolfram Mathematica. We use chain rule and wave transform to convert the equation into integer order nonlinear ordinary differential equation. Hence, we don’t need any discretization, normalization, or reduction. Moreover, 3D graphical representations are given. With the help of these representations, we can have an idea on the physical and geometrical behavior of the solutions.

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

A comparative analysis of Noyes-Field model for the non-linear Belousov–Zhabotinsky reaction using two reliable techniques

Within the domain of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has consistently captivated researchers, sustaining its position as a vibrant and dynamic area of exploration. As an enduring subject of study, the Belousov-Zhabotinsky system provides continuous opportunities to unravel the foundational tenets of nonlinear dynamics within intricate systems. In our quest to deepen comprehension of this complex system, we introduce an innovative methodology for addressing the time fractional Belousov-Zhabotinsky system. This novel approach leverages both the Natural Transform Iterative Method (NTIM) and the Optimal Homotopy Asymptotic Method (OHAM), aiming to contribute novel insights and methodologies to the ongoing discourse surrounding this intriguing and influential area of research. We obtained a series solution to test the accuracy of the suggested approach. The proposed technique has the advantage of requiring few calculations while producing high-precision results. To help you better understand, 2D and 3D graphical representations are provided to demonstrate the model's behavior and how changing the fractional order derivative in Caputo's sense and time affects the solutions.

Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials

This work dives into the Conformable Stochastic Kraenkel-Manna-Merle System (CSKMMS), an important mathematical model for exploring phenomena in ferromagnetic materials. A wide spectrum of stochastic soliton solutions that include hyperbolic, trigonometric and rational functions, is generated using a modified version of Extended Direct Algebraic Method (EDAM) namely r+mEDAM. These stochastic soliton solutions have practical relevance for describing magnetic field behaviour in zero-conductivity ferromagnets. By using Maple to generate 2D and 3D graphical representations, the study analyses how stochastic terms and noise impact these soliton solutions. Finally, this study adds to our knowledge of magnetic field behaviour in ferromagnetic materials by shedding light on the effect of noise on soliton processes inside the CSKMMS.

A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations

<p>Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.</p>

On the extraction of complex behavior of generalized higher-order nonlinear Boussinesq dynamical wave equation and (1+1)-dimensional Van der Waals gas system

<p>In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.</p>

Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels

In the realm of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has long held the fascination of researchers. The Belousov-Zhabotinsky system continues to be an active area of research, offering insights into the fundamental principles of nonlinear dynamics in complex systems. To deepen our understanding of this intricate system, we introduce a pioneering approach to tackle the time fractional Belousov-Zhabotinsky system, employing the Caputo and Atangana-Baleanu Caputo fractional derivatives with the double Laplace method. The solution we obtained is in the form of series which helps in investigating the accuracy of the proposed method. The primary advantage of the proposed technique lies in the low amount of calculations required and produce high degree of precision in the solutions. Furthermore, the existence and uniqueness of the solution are investigated thereby enhancing the overall credibility of our study. To visually represent our results, we present a series of 2D and 3D graphical representations that vividly illustrate the behavior of the model and the impact of changing the fractional order derivative and the time on the obtained solutions.

Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis

In this paper, we explore analytical solutions for the (3+1)-dimensional time-fractional modified Korteweg–de Vries Zakharov-Kuznetsov equation, which incorporates a conformable derivative. Our interest in this model is driven by its significant role in simulating ion-acoustic waves in magnetized plasma. We adopt the unified Riccati equation expansion method and the new Kudrashov method to discover soliton solutions. Our approach uncovers various soliton types, such as kink, singular, periodic-singular, and bright solitons. We conduct a thorough analysis of how different parameters affect wave propagation, enhancing our study with descriptive figures and insightful observations. Furthermore, we delve into the modulation instability characteristic of this model. The influence of specific parameters, like wave number and the order of the conformable derivative, on wave dynamics is demonstrated through detailed visualizations. We also present 2D and 3D graphical representations of these solutions.