Solitary Wave Solutions Research Articles

Generation of Solitary Waves with Analytical Solution for The (3+1)-dimensional pKP-BKP Equation and Reductions

In this study, new solitary wave solutions are obtained for the combination of the B-type Kadomtsev-Petviashvili (BKP) equation and the potential Kadomtsev-Petviashvili (pKP) equation, called the integrable (3+1)-dimensional coupled pKP-BKP equation, and its two reduced forms. For this purpose, the Bernoulli auxiliary equation method, which is an ansatz-based method, is used. As a result, kink, lump, bright soliton and breather wave solutions are observed. It is concluded that this method and the results obtained for the considered pKP -BKP equations are an important step for further studies in this field.

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.

Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation

Abstract In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

Stability analysis and solitary wave solutions for Yu Toda Sasa Fukuyama equation

Stability analysis and solitary wave solutions for Yu Toda Sasa Fukuyama equation

Solitary wave solutions and their limits to the fractional Schrödinger system

This study is concerned with solitary wave solutions and the dynamic behavior of the (2+1)-dimensional nonlinear fractional Schrödinger system. By exploring the dynamic properties of the equilibrium levels to the corresponding Hamiltonian, the expressions of exact solutions of the above system are obtained, including solitary wave solutions, periodic wave solutions, singular periodic wave solutions, singular wave solutions, kink wave solutions, and anti-kink wave solutions. Moreover, the linear stability, geometric characteristics, and limiting behavior of these solutions to the nonlinear fractional Schrödinger system were investigated.

Solitary and traveling wave solutions to nematic liquid crystal equations using Jacobi elliptic functions

In our paper we apply the Jacobi elliptic function (JEF) expansion method to obtain exact solutions to the system of equations governing nematic liquid crystals, a system of high importance in nonlinear optics with numerous physical applications. We obtain solutions that are second-order polynomials in terms of JEFs for both the wave function and the tilt angle of molecular orientation. The solutions differ from previously obtained solutions in including both traveling and solitary wave solutions, with and without chirp. They also include the longitudinal dependence of coefficients in the equations, allowing for the management of both the dispersion and diffraction. Only two parameters of the differential equation need to be defined in terms of other coefficients, providing a wide range of flexibility when it comes to constructing solutions.

Ferroelectric frontiers: Navigating phase portraits, chaos, multistability and sensitivity in thin-film dynamics

This research includes the study of the non-linear dynamics of thin-film ferroelectric materials governed by an equation of wave dynamics within the material. This equation plays a key role in both physics and the study of aqueous flow. The work is planned as examining the symmetries group analysis drops, studying the dynamical system features through bifurcation phase portraits, and carrying dynamic phenomena in chaos theory. Diverse techniques are taken, such as Lyapunov exponent, 2D, and 3D phase portraits, Poincaré maps, time series analysis, and sensitivity to multistability under the different conditions of the initial state. In addition, the study involves using the extended hyperbolic function method to obtain the general analytical solutions via which various kinds of solitary wave solutions are produced including trigonometric and hyperbolic functions and periodic, bright, and singular soliton solutions. These solutions are followed by a list of constraint conditions in the form of equations. Visual data of 2D, 3D, and contour plots are presented, with parameters carefully set to reflect various scenarios. Sensitivity analysis is performed using alternative initial conditions, and stability analysis is demonstrated graphically. To fully grasp the dynamic features of these systems and accurately predict outcomes, it is essential to advance new technologies and methodologies that can further enhance our understanding and predictive capabilities in complex systems.

A plethora of novel solitary wave solutions related to van der Waals equation: a comparative study

In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

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.

Solitary-Wave Solutions of the Fractional Nonlinear Schrödinger Equation: I—Existence and Numerical Generation

The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions from a constrained critical point problem and the application of the concentration-compactness theory. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behavior and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili’s iteration with extrapolation.

Exploring Nonlinear Fractional (2+1)-Dimensional KP–Burgers Model: Derivation and Ion Acoustic Solitary Wave Solution

The purpose of this study is to conduct a complete examination into the fractional (2+1)-dimensional nonlinear KP-Burgers (KP-B) model in the setting of quantum plasma. The main goal is to present a mathematical explanation and derivation for the fractional (2+1)-dimensional nonlinear KP-B model. We used the reductive perturbation technique to obtain the fractional (2+1)-dimensional ion acoustic solitary wave, which leads to the nonlinear KP-B model. To solve the fractional space-time KP-B model, we used the modified sub-equation technique and the extended hyperbolic function methodology. This methodology covers rational, periodic wave, and hyperbolic function solutions. Ion acoustic waves were explored in relation to the effects of ion pressure and an external electric field. The effects of density and fractional order on the properties of a single solution are examined. The plasma system is defined as an unmagnetized epi plasma system composed of relativistic ions, positrons, and nonextensive electrons that can be found in a range of astrophysical and cosmological environments. The solution variables include electron-positron and ion-electron temperature ratios, electron and positron nonextensivity strength, ion kinematic viscosity, positron concentration, and the weakly relativistic streaming factor. The fractional order and associated plasma properties have a considerable influence on the phase velocity of ion acoustic waves. When the fractional order equals one, the obtained results are consistent with known outcomes. This work makes a substantial contribution to our understanding of nonlinear events in quantum plasmas, particularly ion acoustic waves. It provides a solid approach for solution development and analysis, with implications for a variety of astrophysical and cosmological scenarios.

Diverse soliton wave profile assessment to the fractional order nonlinear Landau-Ginzburg-Higgs and coupled Boussinesq-Burger equations

The space–time fractional Landau-Ginzburg-Higgs equation and coupled Boussinesq-Burger equation describe the behavior of nonlinear waves in the tropical and mid-latitude troposphere, exhibiting weak scattering, extended connections, arising from the interactions between equatorial and mid-latitude Rossby waves, fluid flow in dynamic systems, and depicting wave propagation in shallow water. The improved Bernoulli sub-equation function method has been used to achieve new and wide-ranging closed-form solitary wave solutions to the mentioned nonlinear fractional partial differential equations through beta-derivative. A wave transformation is applied to renovate the fractional-order equation into an ordinary differential equation. Some standard wave shapes of multiple soliton type, single soliton, kink shape, double soliton shape type, triple soliton shape, anti-kink shape, and other types of solitons have been established. The more updated software Python is used to display the solutions by using 3D and contour plotlines to describe the physical significances of attained solutions more clearly. The findings of this study are straightforward, adaptable, and quicker to simulate. It has been notable that the improved Bernoulli sub-equation function method is practical, effective, and offers more sophisticated solutions that can help togenerate a large number of wave solutions for various models.

Fuzzy uncertainty modeling of generalized Hirota–Satsuma coupled Korteweg–de Vries equation

This article explores the solitary wave solutions of a generalized Hirota–Satsuma Coupled Korteweg–de Vries (HSCKdV) equation. The HSCKdV equation is a mathematical model that describes certain types of long waves, particularly those found in shallow water. The generalized HSCKdV equation is solved exactly using the Homotopy Perturbation Transform Method (HPTM). By applying this technique, the authors obtain solutions in the form of a convergent power series. These solutions offer an understanding of the characteristics of solitary waves within the domain of shallow water waves. The HSCKdV equation has been solved using the adomian decomposition method, and the results have been compared with those obtained from the HPTM. This comparison demonstrates the effectiveness of the HPTM in solving such nonlinear equations. Further, the HSCKdV equation is extended to a fuzzy version considering the initial condition as a fuzzy parameter. Uncertainty in the initial condition is addressed by representing it using triangular and trapezoidal fuzzy numbers. The generalized fuzzy HSCKdV equation is subsequently tackled using the fuzzy HPTM (FHPTM) providing fuzzy bound solutions. Using the FHPTM, we explain the fuzzy results, highlighting how the solitary wave splits into two solitary waves and noting that the lower and upper bound solutions are interchanged due to negative fuzzy results.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

Analysis of multi-wave solitary solutions of (2+1)-dimensional coupled system of Boiti–Leon–Pempinelli

This work examines the (2+1)-dimensional Boiti–Leon–Pempinelli model, which finds its use in hydrodynamics. This model explains how water waves vary over time in hydrodynamics. We provide new explicit solutions to the generalized (2+1)-dimensional Boiti–Leon–Pempinelli equation by applying the Sardar sub-equation technique. This method is shown to be a reliable and practical tool for solving nonlinear wave equations. Furthermore, different types of solitary wave solutions are constructed: w-shaped, breather waved, chirped, dark, bright, kink, unique, periodic, and more. The results obtained with the variable coefficient Boiti–Leon–Pempinelli equation are stable and different from previous methods. As compared to their constant-coefficient counterparts, the variable-coefficient models are more general here. In the current work, the problem is solved using the Sardar Sub-problem Technique to produce distinct soliton solutions with parameters. Plotting these graphs of the solutions will help you better comprehend the model. The outcomes demonstrate how well the method works to solve nonlinear partial differential equations, which are common in mathematical physics.With the help of this method, we may examine a variety of solutions from significant physical perspectives.

Exploring third-order KdV–SIdV families: Analytical solutions, conservation properties, and phase plane trajectories

This study focuses on the Scale-Invariant (SIdV) families of third-order equations, which are among the few known equations sharing the same solitary wave solution of the KdV equation in the form of sech2. The primary objective is to generalize existing or discover new third-order KdV–SIdV families that feature the analytical sech2 solution of the KdV equation. The study aims to identify all admissible advecting velocity functions ensuring that the KdV–SIdV families accommodate every asymptotically decaying KdV traveling wave with the same wave speed, independent of the advecting velocity. Furthermore, the investigation seeks to delineate their corresponding conserved properties, thereby enhancing our understanding of these intriguing nonlinear partial differential equations. It is shown that the discovered KdV–SIdV families, while sharing the same solitary wave solution, exhibit differing conservation properties. Additionally, a qualitative analysis is conducted using phase plane trajectories to examine the traveling wave solutions and their characteristics for a particular SIdV equation. The emphasis lies in comprehensively exploring potential bifurcations and phase portraits of the vector fields governed by the corresponding system in the parametric space. Through the analysis of distinct phase trajectories across various regions, we derive a new diverse range of solitonic wave solutions, including bell/anti-bell solitary waves, peakon waves, singular waves, and periodic singular waves.

Kink phenomena of the time-space fractional Sharma-Tasso-Olver (STO) equation

Abstract This paper aims to obtain exact solutions of solitary waves for the conformable fractional Sharma-Tasso-Olver (STO) equation which plays an important role in nuclear physics to describe the physical occurrences such as the fission and fusion processes. Solitary waves operate central parts in different areas of study such as electromagnetism, atomic quantum theory, as well as special relativity. By means of sub-ode approach with the aid of the modified fractional Riccati-Bernoulli equation, the exact forms of generalized solitary solution of the fractional (STO) equation are found and specified in hyperbolic, trigonometric, and rational functions. This makes the visualization of the fractional effects and the dynamic behaviors of these solutions in 3D and 2D help in establishing practicality for application of the results. The novel analytical results benefit general engineering and mathematical physics in demonstrating that the proposed employment of the given technique allows solving nonlinear problem analytically. These findings are significant for the progress of wave proceedings in the number of applications.