Periodic Soliton Solutions Research Articles

The dynamical study of fractional complex coupled maccari system in nonlinear optics via two analytical approaches

In this work, the modified auxiliary equation method (MAEM) and the Riccati–Bernoulli sub-ODE method (RBM) are used to investigate the soliton solutions of the fractional complex coupled maccari system (FCCMS). Nonlinear partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilizing a travelling wave transformation, the MAEM, and the RBM. As a result, solutions to hyperbolic, trigonometric and rational functions with unconstrained parameters are obtained. The travelling wave solutions can also be used to generate the solitary wave solutions when the parameters are given particular values. There are several solutions that are modelled for different parameter combinations. We have developed a number of novel solutions, such as the kink, periodic, M-waved, W-shaped, bright soliton, dark soliton, and singular soliton solution. We simulate our figures in Mathematica and provide many 2D and 3D graphs to show how the beta derivative, M-truncated derivative and conformable derivative impacts the analytical solutions of the FCCMS.The results show how effectively the MAEM and RBM work together to extract solitons for fractional-order nonlinear evolution equations in science, technology, and engineering.

On the optical soliton solutions of (2+1)-Biswas-Milovic equation via modified new Kudryashov method

This study includes the examination of optical soliton solutions of the (2+1)-dimensional Biswas-Milovic equation, which is an important equation modeling the soliton behavior in optical fibers, which has been introduced to the literature recently. Since the investigated equation and method are recently introduced and not much works has been done, the bell shape and periodic bright optical soliton solutions have been obtained and interpreted by supporting the 3D, 2D and contour graphics by utilizing the modified new Kudryashov method.

Optical soliton propagation in the Benjamin–Bona–Mahoney–Peregrine equation using two analytical schemes

The purpose of this research is to investigate the nonlinear partial differential equation known as Benjamin–Bona–Mahoney–Peregrine equation. Two versatile techniques, namely the extended (G′G2)-expansion method and modified auxiliary equation method is used to extract the exact solutions of the nonlinear Benjamin–Bona–Mahoney–Peregrine equation. Consequently, a diverse range of novel collections of exact traveling wave solutions are identified such as kink, bright, singular kink and periodic soliton solution. To provide a physical understanding, several of the discovered solutions are shown in figures. The resulting solutions are of the trigonometric, hyperbolic, and rational varieties and the stability analysis are also discussed. In addition to this, the graphical aspects of the produced solutions are investigated. Due to the vital necessity of this aspect, we make use of the suitable parameter values to place a priority on the physical aspects of the provided findings by utilizing 3D and contour charts. The suggested procedures are of great use and can be extended to additional nonlinear evolutionary equations that reflect nonlinear physical frameworks in the nonlinear sciences.

Different solitary wave solutions and bilinear form for modified mixed-KDV equation

This study we handled (1+1)-dimensional modified-mixed KDV (mmKdV) equation for unique soliton solutions. For this purpose Firstly, auto-Bäcklund and Cole-Hopf transformations are created through the homogeneous balance method. Various soliton-like solutions are given by the trigonometric, hyperbolic, and exponential function waves and that leads to the extraction of novel periodic, dark, and bright soliton solutions. Moreover, the bilinear form of the model is found by utilizing the logarithmic transformation. Density, 3-D, and 2-D graphics are displayed to illustrate the dynamics of the acquired solutions. It is determined that the approach employed to obtain analytic solutions to nonlinear partial differential equation is efficient and powerful.

On the dynamics of a dual space–time fractional nonlinear Schrödinger model in optical fibers

The nonlinear Schrödinger equation (NLSE) is a significant nonlinear complex model known as the most important struture to represent light pulse propagation in optical fibers, which plays a key role in demonstrating the proliferation specifically short pulse in optical fibers and have large appliances in telecommunication system and ultra fast signal routing in nonlinear optics. In this study, we look into explicit solutions of a coupled fractional NLSE for a dual space–time, which reflect numerous explicit travelling wave solutions, as well as their physical interpretation. We use the fractional complex transform to obtain multiple exact answers using the unified technique and the extended simple equation method. These exact answers include trigonometric function solutions as well as dark, bright, singular, periodic, and optical soliton solutions. For a set of appropriate parameters, Mathematica is used to represent the dynamics of various wave structures as 3D, 2D, and contour visualizations. These findings show that the proposed methodologies are simple, prolific, and capable of studying the nature of numerous physical models of complex phenomena in modern science.

Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media

This article investigates a new generalized two-dimensional nonlinear wave equation of engineering physics with various applications in the fields of sciences and engineering. In this study, shock wave and solitary wave solutions were secured via the sine–Gordon technique. Moreover, various new group invariants along-side exact classical results of the equation were achieved through the utilization of Lie group theoretic techniques. Some of the solutions are gained with regards to Weierstrass functions, complex soliton, topological kink soliton as well as singular soliton. Besides, several algebraic and other solitary-wave-type solutions are obtained. Wave dynamics of the solutions are plotted to give more physical meanings to the obtained results and have a better knowledge of what the nonlinear wave equation represents in terms of physical phenomena. Application of the secured results in engineering (signal processing), physics (electromagnetism) and complex media are presented.

Novel Soliton Solutions for the (3 + 1)-Dimensional Sakovich Equation Using Different Analytical Methods

In the work that we are doing now, our goal is to find a solution to the (3 + 1)-dimensional Sakovich equation, which is an equation that can be used to characterize the movement of nonlinear waves. This approach sees extensive use across the board in engineering’s many subfields. This new equation can represent more dispersion and nonlinear effects, which allows it to accept more different application scenarios. As a result, it has a wider range of potential applications in the physical world. As a consequence of this, when the two components are combined, they can be used to easily counteract the effects of nonlinearity and dispersion on the integrability of partially differential equations. This equation represents a novel physical paradigm that should be looked into further. We use the following three analytical techniques: the exp − ψ η expansion method is the first one, the second is the G ′ / k G ′ + G + r expansion method, and the Bernoulli sub-ODE technique is the last one. Lastly, the soliton solutions obtained are presented by graphs in two and three dimensions, which present different kinds of solitons such as dark, periodic, exponential, bright, and singular soliton solutions.

Optical Solitons in Fiber Bragg Gratings for Fractional Nonlinear Schrödinger Equation with Generalized Anti-cubic Nonlinearity using Conformable Derivative

This work explores Kink soliton solution, periodic soliton solution, and rational function solutions for the fractional generalized anti-cubic (FGAC) nonlinearity in fiber Bragg gratings (BGs). The rational fractional ((D_ζ^α G)/G)-expansion method is employed in conjunction with the idea of a conformable fractional derivative. Due to its nature, the soliton solution looks to have some restrictions.

Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach

This article describes the infinite conservation law, quasi-periodic wave, breather, lump, and characteristic of integrability via bilinear Bäcklund and lax pairs of the non-autonomous Kadomtsev–Petviashvili (NKP) equation (in the presence of an external force and damping). Bell-polynomial is deduced from Bäcklund transformation from which infinite conservation law for NKP equation is derived. Obtaining well-known quasi-periodic solutions is an important aspect of this investigation. A limiting procedure is used for analyzing the asymptotic behavior of multi-periodic waves. This method proves rigorously that periodic waves tend toward soliton solutions under a small amplitude limit, which results in a relationship between periodic waves and soliton solutions. For the first time, the effect of external force in a quasi-periodic wave is demonstrated explicitly from the numerical understanding. Further, some complicated solution structures of the NKP equation such as lump and breather-type solitons are explored from the bilinear form of the said equation with the appropriate choice of polynomial functions. Important characteristics of lump and breather waves in different forcing backgrounds are graphically described. Finally, the lump wave of the NKP equation is explored from the breather wave in the limiting stage. It is also confirmed from the analytical results of the relevant motions that the velocity, maximum altitude, and interacting natures of the wave quantities are all influenced by the damping and forcing terms.

Localized wave solutions for (2+1)-dimensional third-order Fokou-Kofane-Mohamadou-Yomba equation

Analytical and numerical investigations of localized wave solutions for a nonlinear evolution of shallow water waves with surface tension, which is described by the (2+1)-dimensional third-order Fokou-Kofane-Mohamadou-Yomba (FKMY) equation are performed. Furthermore, we show, mainly by Maple software and the Hirota bilinear method, that appropriated ansatzes can be used to generate new large families of traveling localized structures such as lump, soliton, periodic soliton, quasi-periodic soliton, and quasi-periodic breather solutions. Interactions between those soliton solutions and their dependence on the system physical parameters have been carefully analysed. We show that the soliton solutions of the 2D third-order FKMY display a very rich spectrum of dynamical behaviour when its parameters are varied. The present results could be applicable in explaining the basic features of localized disturbances in many fields of science where the 2D third-order FKMY equation appears.

Integrability and new periodic, kink-antikink and complex optical soliton solutions of [formula omitted]-dimensional variable coefficient DJKM equation for the propagation of nonlinear dispersive waves in inhomogeneous media

This article deals with the (3+1)-dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa equation which describes the behavior of waves as they travel through nonlinear dispersive media. The integrability of this equation has been evaluated by mean of Painlevé property. An auto-Bäcklund transformation of the considered equation is being produced with the aid of Painlevé analysis. To get the analytic solutions of the considered equation, the auto-Bäcklund transformation method is employed. Three new analytical solution families including complex solution have been derived successfully via the auto-Bäcklund transformation method. All the results have been plotted in three-dimension by taking different parameter values and functions. These solutions depict the kink-antikink wave, periodic wave, complex periodic wave and complex kink-antikink wave solutions for the considered equation.

Stationary solutions and dynamical evolution of the nonlocal Gross–Pitaevskii equation in spin-2 Bose–Einstein condensates

In this paper, we investigate the nonlocal Gross–Pitaevskii equation which describes the phenomena of Bose–Einstein condensates under the mean field approximation. Using mean-field theory and adopting separation of variables method, the five-component Gross–Pitaevskii spin–orbit-coupled spin-2 Bose–Einstein condensate model can be reduced to five-component ordinary differential equation, then twelve families of solutions are obtained by the combination of some Jacobi elliptic functions. In particular, when the modulus of Jacobi elliptic function tends to one, a series of bright soliton and dark soliton and their combined solution are obtained. These periodic and hyperbolic soliton solutions are analyzed in detail, the nature and images will be of practical application to provide theoretical guidance for spin systems.

A novel investigation of dark, bright, and periodic soliton solutions for the Kadomtsev-Petviashvili equation via different techniques

A novel investigation of dark, bright, and periodic soliton solutions for the Kadomtsev-Petviashvili equation via different techniques

Application of the newly \(\varphi^{6}\)-model expansion approach to the nonlinear reaction-diffusion equation

In this paper, we use the \(\varphi ^{6}\)-model expansion method to construct the traveling wave solutions for the reaction-diffusion equation. The method of \(\varphi ^{6}\)-model expansion enables the explicit retrieval of a wide variety of solution types, such as bright, singular, periodic, and combined singular soliton solutions. Kink-type solitons, also known as topological solitons in the context of water waves, are another type of solution that can be explicitly retrieved. This study’s results might enhance the equation’s nonlinear dynamical properties. The method proposes a practical and efficient method for solving a sizable class of nonlinear partial differential equations. The dynamical features of the data are explained and highlighted using exciting graphs.

New families of soliton solutions for the (2 + 1)-dimensional nonlinear evolution equation arising in nonlinear optics

This article aims to achieve families of new traveling wave solutions to the (2 + 1)-dimensional nonlinear evolution equation using three powerful analytical techniques: the general form of Kudryashov’s method, the Bernoulli sub-ODE method, and the extended direct algebraic method. Moreover, we use several graphics to highlight our findings. The resulting traveling wave solutions are presented by hyperbolic, trigonometric, rational, and exponential functions. Bright, dark, periodic, and singular soliton solutions are introduced, and the obtained solutions are critical in explaining the wave dynamics in various models. The main findings and graphs indicate that the used approaches are the most appropriate for solving the given model, which generated several deferent solutions. Moreover, the dynamics of the model’s produced solutions may be controlled by adjusting the values of the model’s parameters.

Diverse optical soliton solutions to the Kundu-Mukherjee-Naskar equation via two novel techniques

In this work, the Kundu-Mukherjee-Naskar (KMN) equation that addresses the optical soliton dynamics in (2 +1)-dimensions is studied via two novel techniques, which are the Cole-Hopf transform based method and Wang’s direct mapping method for the first time. Firstly, based on the Cole-Hopf transform, the symbolic computation with the ansatz function schemes are employed to construct the abundant soliton solutions. Some new soliton solutions including the rational breathe, cross-periodic, multi-wave, double exponential form and interaction soliton solutions are obtained. Secondly, Wang’s direct mapping method is adopted to seek for the breather, bright-dark and kinky periodic soliton solutions. Meanwhile, the behaviors of the different soliton solutions are presented in the form of 3-D plot, 2-D contour and 2-D curve. The ideas in this work are expected to provide some new enlightenment for the study of the exact soliton solutions of the PDEs in the optics.

Study of solitary and kink waves, stability analysis, and fractional effect in magnetized plasma

The (3 + 1)-dimensional fractional modified Zakharov Kuznetsov (mZK) equation is one of the nonlinear models to indicate the impact of magnetic fields on weak ion-acoustic waves in plasma; made up of cool and hot electrons. The primary goal of the present study is to use the (m+G′G)-expansion technique to seek the solutions of mZK equation. The solutions are gained in the form of kink, dark, singular periodic and W-type soliton solutions. The influence of the fractional parameter on waveforms has also been examined by representing 2D and 3D graphs for distinct values of fractional-order β. Moreover, we utilize Hamiltonian system properties to confirm the stability of the solution. The (m+G′G)-expansion technique can also be used to examine the nonlinear evolution models being developed in various scientific and technological fields, such as mathematical physics and plasma physics. The soliton solutions attained by using the above technique have not been derived yet.

Soliton solutions of the resonant nonlinear Schrödinger equation using modified auxiliary equation method with three different nonlinearities

The dynamical behavior of the resonant nonlinear Schrödinger equation is investigated in this paper. Resonant nonlinear Schrödinger equation describes the wave propagation in fiber optics. The modified auxiliary equation method is used to extract the soliton solutions of resonant nonlinear Schrödinger equation. The modified auxiliary equation method is novel, stable and efficient exact method. This equation is considered with Kerr law, parabolic law and anti-cubic law of nonlinearities. Many novel soliton solutions such as periodic, dark, bell shaped and singular soliton solutions are extracted using the proposed method. The 3D graphs and 2D contour graphs of retrieved solutions are plotted using symbolic software, Maple. The obtained results containing trigonometric function, hyperbolic function and rational functions are hopped to be beneficial to understand the dynamical framework of the related physical phenomena.

Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation

In this study, we explain the impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation (KMNE). In this regard, some new optical solitons are received via the unified method to the aforesaid equation to explain such impacts of the soliton pulses. The presenting optical solitons are expressed by the dark, periodic, bell, bright, kink, and singular soliton solutions. Taking into account the two impacts help stabilize the soliton pulses during their propagation by generating new dynamics depending upon the nonlinearity and the wave dispersion parameters of the studied equation. All the characteristics of the soliton pulses are exhibited graphically. It is found from the graphical outputs that the soliton profiles are decreasing and increasing with the increase of nonlinearity and dispersion parameters, respectively. The outcomes reveal that the soliton pulses are balanced due to the influences of nonlinearity and wave dispersion parameters of the aforementioned equation. It is mentioned that the impact of wave dispersion and nonlinearity parameters on the soliton pulses has not been discussed in the past. Therefore, the applied method permits the explanation of the various wave dynamics by analyzing the attained soliton solutions in nonlinear optical fibers systems, which can be used for further studies.

Diverse Soliton Structures of the ( 2 + 1 )-Dimensional Nonlinear Electrical Transmission Line Equation

In this work, the ( 2 + 1 )-dimensional nonlinear electrical transmission line equation (NETLE) is investigated by applying three recent technologies, namely, the variational approach, Hamiltonian approach, and energy balance approach. Diverse exact soliton solutions such as the bright, bright-like, kinky bright, bright-dark soliton, and periodic soliton solutions are successfully constructed. The outlines of the different solutions are shown in the form of the 3-D plot with the help of the Wolfram Mathematica. It reveals that the used methods are concise and effective and are expected to provide some inspiration for the study of travelling wave solutions of the PDEs in physics.