Solitary Wave Profiles Research Articles

The fractional solitary wave profiles and dynamical insights with chaos analysis and sensitivity demonstration

This work investigates the dynamics of solitary waves in a thin-film ferroelectric material model, an essential component of optical systems. The study recognizes the limitations of the inverse scattering transform in solving the Cauchy problem for this conformable thin-film ferroelectric material model and applies a new Kudryashov approach and Bernoulli’s equation technique to ordinary differential equations. We apply these methods to the present model and obtain closed form analytical solutions. A variety of traveling wave and soliton solutions are constructed, including kink, bell, bright, and dark soliton solutions. It is acknowledged that the nonlinear evolution equation can be transformed into an ordinary differential equation by applying traveling wave transformations. Suitable physical parameters are chosen to create 2D, 3D, and contour plots that graphically depict the graphical dispersion of obtained fractional soliton solutions. The impact of the conformable fractional parameter is further demonstrated by the graphical representation of solitons propagation. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the system’s total energy in terms of momentum and position. While a chaos analysis describes chaotic, quasi-periodic, and periodic behaviors, a sensitivity analysis demonstrates how responsive the model is to various initial conditions. Due to their dual-purpose nature as nonlinear ferroelectric and dielectric materials, thin ferroelectric films are widely used in modern electrical devices.

Signature of conservation laws and solitary wave solution with different dynamics in Thomas–Fermi plasma: Lie theory

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

Modulation instability and optical wave profiles for the conformable Schrödinger–Poisson dynamical system

This manuscript studies the exact solitary wave profiles for the conformable Schrödinger–Poisson dynamical system. This system has a significant role in gravity’s quantum state operation approximates the interaction between quantum mechanics and gravitation. The diverse exact solitary wave profiles are constructed by using the Khater method. The different closed-form solutions are obtained such as dark, singular, hyperbolic, periodic, and plane waves. Moreover, the modulation instability is also discussed for this dynamical system. To, show the physical significance of desired results we plot the 3D and 2D plots along with the related contour graphs, which are also drawn. The constraint criteria for the exact solutions are also exposed.

Bifurcation, Stability, and Nonlinear Parametric Effects on the Solitary Wave Profile of the Riemann Wave Equation

Bifurcation, Stability, and Nonlinear Parametric Effects on the Solitary Wave Profile of the Riemann Wave Equation

On solitary-wave solutions of Rosenau-type equations

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

The sensitivity demonstration and propagation of hyper-geometric soliton waves in plasma physics of Kairat-II equation

This study investigates the Kairat-II equation, describing optical pulse behavior in optical fibers and plasma. To uncover new solitary wave profiles, the study employs an extended direct algebraic method. This kind of solution has never been reached in research prior to this study. This innovative approach efficiently encompasses a comprehensive set of thirty-seven solitonic wave profiles, spanning various soliton families. The investigation unveils novel solitonic wave patterns, including plane solutions, hyper-geometric solutions, mixed hyperbolic solutions, periodic and mixed periodic solutions, mixed trigonometric solutions, trigonometric solutions, shock solutions, mixed shock singular solutions, mixed singular solutions, complex solitary shock solutions, singular solutions, and shock wave solutions. To demonstrate the pulse propagation characteristics, the research presents 2-D, 3-D, and contour graphics based on parameter values, aiding in a better understanding of the phenomenon.

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.

Ion acoustic solitary waves in multicomponent plasmas: Influence of ion drift velocity and nonthermal electrons

AbstractIon acoustic solitary waves with arbitrary amplitude have been investigated using the Sagdeev potential (SP) method in a collisionless plasma consisting of non‐isothermal electrons, positrons, and positive ions. The SP has been obtained and the dependence of large amplitude solitary structures on various plasma parameters such as ion drift velocity , non‐thermal parameter , electron to positron temperature ratio , positron density , and Mach number have been studied. It is shown that, for the same values of plasma parameters, accounting for ion drift velocity greatly corrects the solitary wave profile.

A new sensitive visualization, solitary wave profiles and conservation laws of ion sound waves arising in plasma

A new sensitive visualization, solitary wave profiles and conservation laws of ion sound waves arising in plasma

Solitary-Wave Solutions of Benjamin–Ono and Other Systems for Internal Waves: II. Dynamics

Considered here are two systems of equations modeling the two-way propagation of long-crested, long-wavelength internal waves along the interface of a two-layer system of fluids in the Benjamin–Ono and the Intermediate Long-Wave regime, respectively. These systems were previously shown to have solitary-wave solutions, decaying to zero algebraically for the Benjamin–Ono system, and exponentially in the Intermediate Long-Wave regime. Several methods to approximate solitary-wave profiles were introduced and analyzed by the authors in Part I of this project. A natural continuation of this previous work, pursued here, is to study the dynamics of the solitary-wave solutions of these systems. This will be done by computational means using a discretization of the periodic initial-value problem. The numerical method used here is a Fourier spectral method for the spatial approximation coupled with a fourth-order, explicit Runge–Kutta time stepping. The resulting, fully discrete scheme is used to study computationally the stability of the solitary waves under small and large perturbations, the collisions of solitary waves, the resolution of initial data into trains of solitary waves, and the formation of dispersive shock waves. Comparisons with related unidirectional models are also undertaken.

Highly nonlinear internal solitary waves with a free surface

A new mathematical model is derived for internal wave propagation in a two-layer fluid domain with a free surface in which the lower layer is of finite depth. The mathematical model consists of three coupled nonlinear equations for displacement of the interface and velocity potentials of the two layers, each of which is expressed as the convex combination of their values at two arbitrary depths. Model equations are correct up to O(μ4) terms, where μ is the ratio of harmonic mean of the depths of the two layers to double of a typical wavelength. Finite amplitude internal solitary wave profiles as well as free surface wave profiles are obtained numerically for different wave speeds, density ratios and depth ratios. Interfacial solitary wave profiles obtained here are in very good agreement with recent experimental observations of Zhao et al. (2020). The figures presented here demonstrate how the interfacial wave profile predicted by the free surface model differs from predictions of rigid-lid two-layer fluid models.

The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures.

New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature.

The First Integral of the Dissipative Nonlinear Schrödinger Equation with Nucci’s Direct Method and Explicit Wave Profile Formation

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1+1)− dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci’s scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci’s direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci’s scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution.

Exact Fractional Solution by Nucci’s Reduction Approach and New Analytical Propagating Optical Soliton Structures in Fiber-Optics

This study examines the Chen–Lee–Liu dynamical equation, which represents the propagation of optical pulses in optical fibers and plasma. A new extended direct algebraic technique and Nucci’s scheme are used to find new solitary wave profiles. The method covers thirty-seven solitonic wave profiles, including approximately all soliton families, in an efficient and generic manner. New solitonic wave patterns are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions. The exact fractional solution is obtained using Nucci’s reduction approach. The impact of the fractional order parameter on the solution is considered using both mathematical expressions and graphical visualization. The fractional order parameter is responsible for controlling the singularity of the solution which is graphically displayed. A sensitivity analysis was used to predict the sensitivity of equations with respect to initial conditions. To demonstrate the pulse propagation characteristics, while taking suitable values for the parameters involved, 2-D, 3-D, and contour graphics of the outcomes achieved are presented. The influence of the fractional order ζ is shown graphically. A periodic-singular wave with lower amplitude and dark-singular behaviour is inferred from the graphical behaviour of the trigonometric function solution H1 and the rational function solution H34 from the obtained solutions, respectively.

Effects of vorticity on solitary waves

The vorticity effect on solitary wave profiles has not been solved experimentally; previous studies theoretically and numerically showed that when a solitary wave progressed in the positive direction, the effective wavelength of a solitary wave with positive vorticity increased. Using laboratory experiments and fully nonlinear numerical simulations, we here show that the effective wavelength is extended more when positive vorticity is given to a progressive wave in the positive direction. We further show that the total energy increases with increasing positive vorticity, demonstrating that a wave with positive vorticity propagates with less attenuation and lasts longer than a solitary wave with no vorticity. We anticipate that our outcomes will provide a starting point for more sophisticated methods to investigate the effect of vorticity on solitary waves in laboratory experiments and numerical simulations.

The fractional wave propagation, dynamical investigation, and sensitive visualization of the continuum isotropic bi-quadratic Heisenberg spin chain process

This paper deals with the Lakshmanan-Porsezian-Daniel equation which delineates the continuum isotropic bi-quadratic Heisenberg spin chain phenomenon. A new auxiliary equation method is exerted on the considered equation to find solitary wave profiles. It is a simple and powerful approach for developing innovative wave profiles based on diverse soliton families such as trigonometric functions, rational, hyperbolic trigonometric function and exponential function etc. As a result, the solitonic wave patterns attain such as dark, bright, dark-bright, singular, rational, periodic-singular, exponential, and periodic solitons etc. The deep dynamical aspects of the governing model study by performing the chaos and sensitivity analysis. The planer dynamical system of equation develop and satisfy the Hamiltonian criteria to assure that, the developed system is Hamiltonian dynamical system and contains all traveling wave structures and the system is conservative. The graphical explanation of energy levels presents the significant insights and the existence of closed-form solutions to the model. The periodic, quasi-periodic, and quasi-periodic-chaotic profiles are present to see the deep dynamics of the continuum isotropic bi-quadratic Heisenberg spin chain system. The graphically visualization for sensitivity analysis of the governing equation portraits by taking some initial values to verify its dependence. It is shown that, the model is more sensitive regarding to initial conditions rather then parameters. The graphical two dimensional, three dimensional, and contour visualization of the obtained results are presented to express the pulse propagation behavior by assuming the appropriate values of the involved parameters. The impact of fractional parameter is displayed in the graphical sense. The fractional order controls the soliton behaviour which means that, the prediction and precautions can be constructed about the physical phenomenon of the continuum isotropic bi-quadratic Heisenberg spin chain. As a results, the fractional order exhibits the states of distortion in continuum bi-quadratic magnetic system with non-zero vector on which the form evaluates to zero. The graphical two dimensional, three dimensional, and contour visualization of the obtained results are presented to express the pulse propagation behavior by assuming the appropriate values of the involved parameters.

Solitary Wave Interaction with a Floating Pontoon Based on Boussinesq Model and CFD-Based Simulations

A mathematical model of solitary wave interaction with a pontoon-type rigid floating structure over a flat bottom is formulated based on Boussinesq-type equations under weakly nonlinear dispersive waves. Based on the higher-order Boussinesq equations, the solitary wave equation is derived, and a semi-analytical solution is obtained using the perturbation technique. On the other hand, brief descriptions of the application of wave2Foam and OceanWave3D on the aforementioned problem are presented. The analytical solitary wave profiles in the outer region are compared with Computational Fluid Dynamics (CFD) and OceanWave 3D model simulations in different cases. The comparison shows a good level of agreement between analytical, wave2Foam, and OceanWave3D. In addition, based on the wave2Foam and coupled OceanWave3D model, the horizontal, vertical wave forces, and the pressure distributions around the pontoon are analysed. Further, the effect of the Ursell number, pontoon length, and water depth on the solitary wave profiles are analysed based on the analytical solution. The paper validates each of the three models and performs intercomparison among them to assess their fidelity and computational burden.

Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model

In the past few decades, huge surges and waves generated during tsunami events have caused devastating destruction to both onshore and offshore infrastructures, seriously threatening the safety and intactness of coastal communities around the world. As one of the most important coastal protection infrastructures, a seawall can effectively reduce the impact intensity of surges and waves, thus playing a vital role in protecting coastal regions. Most previous studies have systematically investigated the hydrodynamic characteristics of seawalls under the action of regular and irregular waves. Meanwhile, solitary wave models are often used as the wave model for tsunamis. However, vast hydrodynamic differences exist between solitary wave models and real-world tsunamis in terms of both wave profile and wave period. Hence, our understanding of the performance of seawalls in mitigating the damaging power of real-world tsunamis is still insufficient. Hence, it is of great significance to systematically study the performance of seawalls in mitigating the huge surges and waves generated during real-world tsunamis. In comparison to previous research, where the wave profiles of solitary waves were used as tsunami wave models, a parameterized tsunami-like wave based on the recorded wave profile of the 2011 Japan tsunami is applied in this study to evaluate the performance of different seawalls in mitigating tsunami surges and waves by using a nonhydrostatic numerical wave model (NHWAVE). The effects of the prominent factors, such as incident wave height, still water depth, beach slope, and the side slope of the seawall, on the hydrodynamics of the seawall are systematically discussed. It is believed that the research findings drawn from this study can further broaden our understanding of the performance of seawalls in mitigating tsunamis.