Soliton Wave Solutions Research Articles

Newly exploring the Lax pair, bilinear form, bilinear Bäcklund transformation through binary Bell polynomials, and analytical solutions for the (2 + 1)-dimensional generalized Hirota–Satsuma–Ito equation

The (2+1)-dimensional generalized Hirota–Satsuma–Ito equation describing the numerous wave dynamics in shallow waters is investigated in this study. The integrable characteristics of the aforesaid equation, such as a bilinear Bäcklund transformation and Lax pair, are revealed using the Bell polynomials method. First, using this technique, with the aid of Hirota operators, the bilinear form is constructed for the considered equation. In addition, the bilinear Bäcklund transformation and the Lax pair of the aforesaid equation are derived successfully using the bilinear form. Moreover, the bilinear form is also used to construct analytical solutions utilizing the three-wave approach with a test function. While using this method, numerous analytical solutions are derived, which are not presented in the literature. A three-dimensional graph has been plotted for each of the obtained results by giving the appropriate values of the free parameters. These plots reveal a wide variety of wave behavior, such as kink-soliton, periodic wave, anti-kink soliton, and complex periodic wave solutions.

Highly dispersive optical solitons and solitary wave solutions for the (2 + 1)-dimensional Mel’nikov equation in modeling interaction of long waves with short wave packets in two dimensions

In this paper, the optical soliton and solitary wave solutions of the [Formula: see text]-dimensional Mel’nikov equation are investigated using the Kudryashov [Formula: see text] function technique. The Kudryashov [Formula: see text] function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the [Formula: see text]-dimensional Mel’nikov equation have been displayed in 3D, 2D and contour graphs.

Computational simulations of propagation of a tsunami wave across the ocean

The 2D regularized long-wave equation is a mathematical model used to describe the behavior of water waves in two dimensions, taking into account nonlinear and dispersive effects. To obtain soliton wave solutions, computational methods have been employed to solve this equation. Solitons are localized wave solutions that maintain their shape and speed over long distances, making them useful in various applications, such as optical fiber communication and modeling water waves. This study utilized the Khater II, septic-B-spline, and Adomian decomposition techniques to solve the 2D regularized long-wave equation and obtain soliton wave solutions. The results showed a wide range of soliton solutions, including bright and dark solitons as well as soliton pairs and chains. Furthermore, we investigated the impact of the regularization parameter on the soliton solutions and observed that it affects the soliton speed and shape. Our findings demonstrate how computers can be utilized to obtain new soliton wave solutions for the 2D regularized long-wave equation, which have implications for various fields, such as oceanography, engineering, and physics. By understanding how solitons behave within the context of this equation, we can create more accurate models of water waves and other physical systems.

On the exploration of soliton solutions of the nonlinear Manakov system and its sensitivity analysis

In this study, we investigate the fractional coupled nonlinear Schrödinger equation (FCNLSE) of Manakov type, which has numerous applications in different fields of physics, such as optics and plasma physics. In this study, we employ two different analytical methods, namely the auxiliary equation method (AEM) and the improved F-expansion method, to construct novel analytic exact solitary wave solutions, encompassing trigonometric, hyperbolic, and exponential functions. These solutions include solitary wave solutions dark, bright, combo, singular, rational form solutions, and periodic wave solutions. Additionally, using the characterizations of the Hamiltonian system, the stability property of the discovered soliton wave solutions is examined. We have also illustrated the sensitivity analysis for the modified dynamic structural system using different initial conditions. In order to illustrate various physical properties of wave structures, the well-furnished results are finally demonstrated in various 3D, 2D, contour, and density profiles. This work’s findings highlight the significance of researching diverse nonlinear wave phenomena in physics and nonlinear optics by demonstrating how crucial it is to comprehend the physical meaning and behavior of the examined equation. The procedures used are capable, effective, and succinct enough to make it possible for further research.

Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE

Under investigation in this paper is the cubic–quintic nonlinear Schrödinger equation, which describes the propagation of optical on resonant-frequency fields in the inhomogeneous fiber. According to abundant previous researches on the model, exact soliton solutions and rogue wave solutions have been derived through Darboux transformation. The modulation instability phenomenon has been analyzed to evaluate the ability of an initially perturbed plane wave to split into localized energy packets when propagating in a dispersive and nonlinear medium.Numerical solutions with high accuracy are needed in fields of production and engineering. Nonetheless, the data acquisition costs of the optical pulse transmission system is high, which will limit the accuracy and the efficiency of typical numerical and data-driven methods. With the physical knowledge embedded into neural networks in the form of loss function, the problem of big data dependence has been solved. For dynamic analysis on optical pulses with small amount of known information, we strive to obtain high accuracy numerical solutions. Considering the case that the cubic–quintic nonlinear Schrödinger equation is converted to the Kundu–Eckhaus equation with simplified coefficient constraints through variable transformation, we construct modified physics-informed neural networks, where conversions on the input and output are attached to deep neural networks. Training networks with the given initial and boundary data, we effectively derive the expected soliton and rogue wave solutions, where the approximated one-soliton, two-soliton, first-order and second-order rogue waves are included. In general, the modified network reaches low prediction errors with small data available. With the coefficients of equations, the weights and the bias of networks combined as parameters to be trained, we deduce the corresponding value of condition settings for different systems. Moreover, we simulate diverse localized waves in the context of nonlinear electrical transmissions with different environment settings and compare the evolution process to reach conclusions on the parameter discovery.

New Exact Wave Solutions on the Complex Ginzburg–Landau Equation with Extended Rational Sin–Cos and Sinh–Cosh Method

In this paper, a new complex waves representing solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity is investigated. we used the extended rational sin–cos and sinh–cosh function methods construct precise solutions to the nonlinear equation. Bright periodic solution, periodic solution, dark wave soliton solution of phenomena that occur in nonlinear optics or in plasma physics are obtained. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. The proposed method provides an important and effective mathematical tool to construct exact solutions according to different complex equations. The results present the dynamics properties of the different waves with different the 3D and contour plots.

A family of solutions of the time–space fractional longitudinal wave equation

In this article, we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative. Through the soliton ansatz method and a direct integration approach with the symmetry condition, new soliton and solitary wave solutions are derived. Furthermore, the existing conditions of these obtained solutions are also given in this text. These new results add to the existing literature. We believe that they can provide a new window into the understanding of this model.

Analytical and numerical investigation of soliton wave solutions in the fifth-order KdV equation within the KdV-KP framework

This study investigates the fifth-order Korteweg–de Vries (fKdV) equation, which is a generalized form of the classical KdV equation governing weakly nonlinear waves in dispersive media. By employing analytical and numerical techniques, the Khater II and variational iteration methods are utilized to obtain accurate soliton wave solutions. The combination of these approaches ensures the reliability and precision of the analytical outcomes. Graphical representations, including two-dimensional, three-dimensional, and contour plots, visually illustrate the results. Additionally, the stability of the constructed solutions is assessed through the characterization of the Hamiltonian system, providing valuable insights into the soliton wave solutions’ stability properties. This research integrates analytical and numerical methods to explore the fKdV equation within the KdV-KP framework, offering novel soliton wave solutions and enhancing their interpretation through graphical representations while assessing their stability using the Hamiltonian system’s characterization.

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Periodic bright–dark soliton, breather-like wave and rogue wave solutions to a $${\bar{p}}$$-GBS equation in (3+1)-dimensions

Periodic bright–dark soliton, breather-like wave and rogue wave solutions to a $${\bar{p}}$$-GBS equation in (3+1)-dimensions

Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model

The traveling wave solutions and soliton solutions of the conformable M-fractional modified nonlinear Schrödinger equation have been devised in this work by employing the generalized projective Riccati equations technique. The proposed method is straightforward and concise to obtain exact soliton solutions of nonlinear partial differential equations in various scientific discipline. This approach has been applied successfully to a wide variety of physical and mathematical problems. In many cases, the method has been able to provide new insights into the behavior of these systems and has led to the discovery of new phenomena. Several types of soliton solutions, including hyperbolic traveling wave solution, kink-shape soliton solution, singular soliton solution and dark soliton-type traveling wave solution, have been derived from the governing equation by using this effective technique. The solutions achieved by the governing model contain significant and dynamic justifications for certain real-world physical phenomena. Some acquired soliton solutions are displayed in 3D plots, corresponding contour plots and density plots to illustrate the physical dynamics of the governing model. Furthermore, the resulting outcomes manifest that the proposed method is highly efficient for revealing wave solutions for a wide range of nonlinear partial differential equations that emerge in optics.

Hodograph transformation, various exact solutions and dynamical analysis for the complex Wadati–Konno–Ichikawa-II equation

This paper is devoted to studying the second type of complete integrable complex Wadati–Konno–Ichikawa-II (cWKI-II) equation. We first study the conservation laws and construct the hodograph transformation for this equation. Based on the resulting transformation, we construct a new two-component associated cWKI-II equation, which is completely equivalent to the cWKI-II equation. Based on the corresponding Lax pair for this new associated equation, we construct its N-fold Darboux transformation and generalized (m,N−m)-fold Darboux transformation. Then, the associated Bäcklund transformation is used to give various exact solutions, such as smooth soliton, bursting soliton, singular-loop soliton, position controllable semi-rational soliton, breather and rogue wave solutions. The soliton structures along with dynamical evolution behaviors are discussed and shown graphically. Moreover, we investigate some important physical quantities of some soliton solutions such as amplitude, wave number, wave width, velocity, energy and phase by using the asymptotic analysis technique. Finally, we also summarize the formation reasons for some singular structures under some parameter constraints.

New optical soliton solutions for time-fractional Kudryashov’s equation in optical fiber

In this paper, the new Kudryashov approach is considered to find several new exact solutions to time-fractional Kudryashov’s equation which describes the propagation pulses in optical fibers. The present novel optical solutions are expressed via the exponential and hyperbolic functions which are categorized as dark-bright, dark, bright, bell-shape, wave, and singular optical soliton solutions. The two-dimension, three-dimension, and cantor graphs of the dark, bright, bell-shape, wave, and singular solutions are depicted to illustrate the magnitude of the time-fractional Kudryashov’s equation choosing suitable values of physical parameters. Furthermore, the effect of the parameter of time and the conformable fractional order derivative is also presented. Finally, the new Kudryashov approach is an efficient technique to analysis the analytic solutions of the differential equations of integer and fractional orders.

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

In this work, we perform a comprehensive analytical study to find the novel exact traveling wave solutions of the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The recently developed (G′G′+G+A) -expansion technique is a capable method for finding the new exact solutions of assorted nonlinear evolution equations. Some new analytical solutions are obtained by utilizing the aforementioned method. The obtained solutions are expressed as trigonometric functions and exponential functions. The extracted exact wave solutions are advanced and fully unique from the earlier literature Moreover, we have presented the contour simulations, 2D and 3D graphical representations of the solution functions and we have observed that the solutions obtained are periodic and solitary wave solutions. We have shown two soliton wave solutions and two singular periodic wave solutions for the particular values of the parameters graphically. As per our knowledge, we must say that the extracted solutions might be significant and essential for new physical phenomenon.

Exact solutions and Darboux transformation for the reverse space–time nonlocal Lakshmanan–Porsezian–Daniel equation

The integrable reverse space–time nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation is proposed by nonlocal reduction of the AKNS scattering problem. The 1-fold and N-fold Darboux transformations of the nonlocal LPD equation are constructed, from which some interesting nonlinear wave solutions including soliton and rogue wave solutions are obtained, whose dynamical properties are discussed by evolution plots. It is worth mentioning that the real and imaginary parts of the soliton solutions show the characteristics of the breathing solution. These results may be useful in nonlinear optics and related fields.

Various Solitons and Other Wave Solutions to the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods. Abundant exact solutions such as the bright soliton, dark soliton, combined bright–dark soliton, singular soliton and other periodic wave solutions expressed by the generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions are obtained. The numerical results are illustrated in the form of 3D plots, 2D contours and 2D curves by choosing proper parametric values to interpret the physical behavior of the model. The obtained results in this work are expected to provide a rich platform for constructing the soliton solutions of PDEs in physics.

Exact solutions and Darboux transformation for the reverse space–time non-local fifth-order non-linear Schrödinger equation

In this paper, the non-local reverse space−time fifth-order non-linear Schrödinger(NLS) equation has been investigated, which is proposed by the non-local reduction of Ablowitz–Kaup–Newell–Segur (AKNS) scattering problems. The determinant representation of the Nth Darboux transformation for the non-local reverse space−time fifth-order NLS equation is obtained. Some interesting non-linear wave solutions, including soliton, complexiton, and rogue wave solutions, are derived by the Darboux transformation. Moreover, the dynamics of non-linear wave solutions are illustrated with the corresponding evolution plots, and the results show that the non-local fifth-order NLS equation has new different properties from the local case.

Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method

Most of the nonlinear phenomena are described by partial differential equation in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology and mathematical finance. The solutions of a wide range of nonlinear evolution equations exhibit the wave behavior corresponding to the underlying physical systems. In particular, solitary wave solutions and soliton solutions are of great interest for researchers owing to many applications in different areas of science. The behavior of gas diffusion in a homogeneous medium is described by the (2+1)-dimensional Chaffee–Infante equation. In the present study, the modified Khater method is used to solve (2+1)-dimensional Chaffee–Infante equation because it provides various forms of solitons. The bright, dark, and periodic traveling wave patterns are produced by choosing different values of parameters. The solutions are presented graphically through 2D, 3D and contour graphs. The obtained solutions are demonstrated that the modified Khater method is a more useful tool than the existing techniques for solving such nonlinear problems.

Incorporating fractional operators into interaction dynamics studies: An eco-epidemiological model

Analytical solutions are the most effective tools for understanding phenomena modeled with differential equations. On the other hand, analytical solvers may be limited in their ability to handle complex problems and we are forced to rely on approximate and numerical methods. Through the investigation of two types of fractional operators, this article aims to investigate both approaches to studying the interactive dynamics of an eco-epidemiological diffusion model. We consider two differential operators in our main model: local beta time and non-local Riesz space fractional derivative. Firstly, we propose novel classes of soliton wave solutions for the local time fractional version of the system using two efficient analytical methods. In the second part of the paper, we consider the case where the second-order space derivative operator in the problem is of the type of Riesz fractional operator that is non-local. A finite difference scheme is the main idea behind discretizing this fractional system. Throughout this article, we highlight the validity and efficiency of both analytical and numerical perspectives. The results can be examined by epidemiologists for a more comprehensive interpretation.

Novel soliton wave solutions of a special model of the nonlinear Schrödinger equations with mixed derivatives

In this study, the influence of the integrability requirement on nonlinear Schrödinger equations with mixed derivatives is examined. The Rangwala–Rao (RR) equation is named after A. Rangwala, who in 1990 was the first to quantify these effects in their entirety. The aim of our research is identifying how to generate individual waves and how they interact with one another. This is to get a better understanding of the dispersion effect and the progressive variation of the electric field envelope during pulse propagation in optical fibers. The direct algebraic method (DAM), the extended direct algebraic approach (exDAA), and the extended Riccati expansion technique (exREM) are used to develop unique solitary wave solutions for the model under consideration. These numerically calculated solutions illustrate the dynamic behavior of an optical pulse traveling through a cable. Comparing our results to those of other researchers demonstrates the originality of our work.