Search Results (219)

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

This study investigates the potential Kadomtsev–Petviashvili equation incorporating a power-type nonlinearity (PKPp), a model that features prominently in various nonlinear phenomena encountered in physics and applied mathematics. A complete Noether symmetry classification of the PKPp equation is conducted, revealing four distinct scenarios based on different values of the exponent $p$, namely, the general case where $p\ne 1,-1,-2,$ and three special cases where $p=1,$$p=-1,$ and $p=-2.$ Corresponding to each case, conservation laws are derived through a second-order Lagrangian framework. Furthermore, Lie group analysis is employed to reduce the nonlinear partial differential Equation (NLPDE) to ordinary differential Equations (ODEs), thereby enabling the effective application of the Kudryashov method and direct integration techniques to construct exact solutions. In particular, exact solutions of of the considered nonlinear partial differential equation are obtained for the cases $p=1$ and $p=2,$ illustrating the practical implementation of the proposed approach. The solutions obtained include solitary wave, periodic, and rational-type solutions. These results enhance the analytical understanding of the PKPp equation and contribute to the broader theory of nonlinear dispersive equations. Full article

In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/G, 1/G)-expansion approach, we derive different categories of exact solutions, like hyperbolic, trigonometric, and rational functions. The beta fractional derivative is used here to generalize the classical idea of the derivative, which preserves important principles. The derived solutions with broader nonlinear wave structures are periodic waves, breathers, peakons, W-shaped solitons, and singular solitons, which enhance our understanding of nonlinear wave dynamics. In relation to these results, the findings are described by showing the solitons’ physical behaviors, their stabilities, and dispersions under fractional parameters in the form of contour plots and 2D and 3D graphs. Comparisons with earlier studies underscore the originality and consistency of the (G′/G, 1/G)-expansion approach in addressing fractional-order evolution equations. It contributes new solutions to analytical problems of fractional nonlinear integrable systems and helps understand the systems’ dynamic behavior in a wider scope of applications. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

The remote sensing inversion of internal solitary waves (ISWs) enables the retrieval of ISW parameters and facilitates the analysis of their spatial variability. In this study, we utilize continuous optical imagery from the FY-4B satellite to extract real-time ISW propagation speeds throughout their evolution from generation to shoaling. ISW parameters are retrieved in the northern South China Sea based on the quantitative relationship between sea surface current divergence and ISW surface features in optical imagery. The inversion method employs a fully nonlinear equation with continuous stratification to account for the strongly nonlinear nature of ISWs and uses the propagation speed extracted from continuous imagery as a constraint to determine a unique solution. The results show that as ISWs propagate from deep to shallow waters in the northern South China Sea, their statistically averaged amplitude initially increases and then decreases, while their propagation speed continuously decreases with decreasing depth. The inversion results are consistent with previous in situ observations. Furthermore, a three-day consecutive remote sensing tracking analysis of the same ISW revealed that the spatial variation in its parameters aligned well with the abovementioned statistical results. The findings provide an effective inversion approach and supporting datasets for extensive ISW monitoring. Full article

This research explores the dynamic characteristics of the soliton neuron model, a mathematical approach used to describe various complicated processes in neuroscience, including the unclear mechanisms of numerous anesthetics. An appropriate wave transformation converts the neuron model into a two-dimensional dynamical system, which takes the form of a conservative Hamiltonian system with a single degree of freedom. This study utilizes qualitative methods from planar integrable systems theory to analyze and interpret phase portraits. The conditions under which periodic, super-periodic, and solitary wave solutions exist are clearly defined and organized into theorems. These solutions are obtained analytically, with several examples depicted through 2D- and 3D-dimensional graphical illustrations. The research also examines how key physical parameters, such as frequency and sound velocity, affect the nature of these solutions, specifically on the width and the amplitude of those solutions. In addition, by inserting a generalized periodic external force, the model exhibits quasi-periodic and chaotic dynamics. These complicated dynamics are visualized using 2D and 3D phase portraits and time series plots. To further assess chaotic behavior, Lyapunov exponents are calculated. Numerical results indicate that the system’s overall behavior is strongly impacted by changes in the external force’s frequency and amplitude. Full article

This review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type evolution equations up to the fifth order, demonstrating their ability to obtain one- and two-soliton solutions, as well as other solitary waves with high accuracy. To provide benchmark solutions for training PINNs, we employ analytical methods such as the nonisospectral generalization of the AKNS scheme of the inverse scattering transform and the auto-Bäcklund transformation. Finally, we discuss recent advancements in PINN methodology, including improvements in network architecture and optimization techniques. Full article

The generation and propagation of water waves in a numerical wave flume with Ursell numbers (Ur) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stokes equations, along with nonlinear free surface boundary conditions and upstream boundary conditions at the wavemaker, were solved to build the numerical wave flume. The generated waves included small-amplitude, finite-amplitude, cnoidal, and solitary waves. For computational efficiency, the Jacobi elliptic function representing the surface elevation of a cnoidal wave was expressed as a Fourier series expansion. The accuracy of the generated waveforms and associated flow fields was validated through comparison with theoretical solutions. For $Ur<26.32$, small-amplitude waves generated using Goring and Raichlen’s wave generation theory matched those obtained from linear wave theory, while finite-amplitude waves matched those obtained using Madsen’s wave generation theory. For $Ur>26.32$, nonlinear wave generated using Goring and Raichlen’s theory remained permanent, whereas that generated using Madsen’s theory did not. The evolution of a cnoidal wave train with $Ur=43.81$ was examined, and it was found that, after an extended propagation period, the leading waves in the wave train evolved into a series of solitary waves, with the tallest wave positioned at the front. Full article

The objective of this manuscript is to investigate the (2+1)-dimensional Chiral nonlinear Schrödinger equation (CNLSE). We employ the traveling wave transformation to convert the nonlinear partial differential equation (NLPDE) into the nonlinear ordinary differential equation (NLODE). Utilizing the two new vital techniques to derive the solitary wave solutions, the generalized Arnous method and the Riccati equation method, we obtained various types of waves like bright solitons, dark solitons, and periodic wave solutions. Sensitivity analysis is also discussed using different initial conditions. Sensitivity analysis refers to the study of how the solutions of the equations respond to changes in the parameters or initial conditions. It involves assessing the impact of variations in these factors on the behavior and properties of the solutions. To better comprehend the physical consequences of these solutions, we showcase them through different visual depictions like 3D, 2D, and contour plots. The findings of this study are original and hold significant value for the future exploration of the equation, offering valuable directions for researchers to deepen knowledge on the subject. Full article

In this research, we investigate the phenomenon of multistability and complex dynamic behaviors in plasma waves by utilizing advanced mathematical techniques. We examine how fractional-order derivatives influence plasma wave stability by applying the fractional diffusion–reaction model, the framework of nonlinear dynamical systems, and the $\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)$ method. The principal direction of our work is associated with different forms of oscillations in the plasma wave: non-linear periodic, solitons, and kink waves. This leads to the study of small amplitude pulses and solitary waves, which are significant in plasma activities. Using bifurcation analysis, we discuss how these waves appear and develop under different conditions, as well as determine which conditions generate the chaotic behavior or highly complex patterns of waves. We study the details of transitions between waves and their chaotic behavior to characterize the laws that govern their plasma environment. Moreover, we have used non-linear modeling and numerical simulations to understand in detail the complex patterns and the factors of stability underlying the phenomena of plasma waves. In addition, our study also investigates the correspondence between non-linearity, multi-stability, and the birth of complex structures such as solitons and kink waves. The solutions of the dynamical system produced by the proposed nonlinear model generate different patterns of response based on system parameter variation. These patterns include oscillations and decay behaviors. Research results about system stability and solution convergence under various parameter settings provide an extended performance evaluation of the proposed method through a better understanding of system dynamics. They increase our understanding of chaotic behavior in plasma systems and pave the way for applications in plasma physics and energy systems, as well as advanced technologies. Full article

In this paper, an equation describing nonlinear wave phenomena on the surface of magnetically active plasma in the approximation of the complete homogeneity of processes along the direction of the constant magnetic field is obtained. One of its solutions, in the form of a pulse having the shape of rapidly decaying oscillations with a changing period, is found to essentially depend on the magnitude of the magnetic field and shown to be approximately described by a specially selected analytical function. A detailed analytical analysis of the properties of another solitary wave formation existing under conditions of resonant coincidence of its carrier frequency with the corresponding value of its eigen surface oscillations in the considered cold semi-infinite plasma, in which a constant magnetic field is directed along its boundary, is also carried out. The conditions for the excitation of wave disturbances are determined, and analytical expressions that adequately describe the space–time structure of nonlinear waves are proposed. Full article

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it is demonstrated that the gBBKW equations are solvable through the consistent Riccati expansion method. Leveraging this property, a novel Bäcklund transformation, solitary wave solution, and soliton–cnoidal wave solution are derived. Furthermore, miscellaneous novel solutions of gBBKW equations are obtained using the modified Sardar sub-equation method. The impact of variations in the diffusion power parameter on wave velocity and height is quantitatively analyzed. The exact solutions of gBBKW equations provide precise description of propagation characteristics for a deeper understanding and the prediction of some ocean wave phenomena. Full article

This study sought to deepen our understanding of the dynamical properties of the newly extended $(3+1)$-dimensional integrable Kadomtsev–Petviashvili (KP) equation, which models the behavior of ion acoustic waves in plasmas and nonlinear optics. This paper aimed to perform Lie symmetry analysis and derive lump, breather, and soliton solutions using the extended hyperbolic function method and the generalized logistic equation method. It also analyzed the dynamical system using chaos detection techniques such as the Lyapunov exponent, return maps, and the fractal dimension. Initially, we focused on constructing lump and breather soliton solutions by employing Hirota’s bilinear method. Secondly, employing Lie symmetry analysis, symmetry generators were utilized to satisfy the Lie invariance conditions. This approach revealed a seven-dimensional Lie algebra for the extended $(3+1)$-dimensional integrable KP equation, incorporating translational symmetry (including dilation or scaling) as well as translations in space and time, which were linked to the conservation of energy. The analysis demonstrated that this formed an optimal sub-algebraic system via similarity reductions. Subsequently, a wave transformation method was applied to reduce the governing system to ordinary differential equations, yielding a wide array of exact solitary wave solutions. The extended hyperbolic function method and the generalized logistic equation method were employed to solve the ordinary differential equations and explore closed-form analytical solitary wave solutions for the diffusive system under consideration. Among the results obtained were various soliton solutions. When plotting the results of all the solutions, we obtained bright, dark, kink, anti-kink, peak, and periodic wave structures. The outcomes are illustrated using 2D, 3D, and contour plots. Finally, upon introducing the perturbation term, the system’s behavior was analyzed using chaos detection techniques such as the Lyapunov exponent, return maps, and the fractal dimension. The results contribute to a deeper understanding of the dynamic properties of the extended KP equation in fluid mechanics. Full article

This study uses a conformable derivative of order $\beta$ to investigate a fractional Whitham–Broer–Kaup ($\mathcal{FWBK}$) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation $\mathcal{EMSSE}$ approach is applied to achieve precise analytical solutions, demonstrating its effectiveness in resolving complex wave photons. Bright, solitary, trigonometric, dark, and plane waves are among the various wave dynamics that may be effectively and precisely determined using the $\mathcal{FWBK}$ model. Furthermore, the study explores the chaotic behaviour of both perturbed and unperturbed systems, revealing illumination on their dynamic characteristics. By demonstrating its validity in examining wave propagation in nonlinear fractional systems, the effectiveness and reliability of the suggested method in fractional modelling are confirmed through thorough investigation. Full article