Dark-bright Soliton Solutions Research Articles

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

In this study, we aim to obtain numerical results of the modified Benjamin-Bona-Mahony equation, Ostrovsky-Benjamin-Bona-Mahony equation and Mikhailov-Novikov-Wang equation via the physics-informed neural networks (PINN) method. The equations are modeled for shallow and long water waves, as well as fundamental and phenomenonal models in ocean engineering. According to the implementation, we obtained the PINN solutions of kink, bright, multisoliton (two-soliton) and mixed dark-bright soliton solutions. According to the inference from the obtained results, we achieved good results in some cases compared to other approximate solution methods in the literature. However, it was also observed that the best possible results could not be obtained in cases where the soliton type was intricate and layered. While the results were obtained, the number of hidden layers and the number of neural networks in the layers also varied. These results are shown in tables. Since it is known that the aforementioned models are not solved by the PINN method, we anticipate that the study will lead to other studies in the field of ocean engineering.

The influence of fractionality and unconstrained parameters on mathematical and graphical analysis of the time fractional phi-four model

In the framework of several nonlinear physical phenomena arising from water wave mechanics, this work recovers some new precise solutions to the time-fractional phi-four equation. For this, the phi-four equation's space and time fractional transformation is converted into an ordinary differential equation (ODE). The improved modified extended tanh function (imETF) approach is then employed by the ODE as a powerful method based on the conformable derivative. Thus, lump solutions, periodic solutions, solitary and multiple soliton solutions, dark-bright soliton solutions, and Jacobi elliptic doubly periodic type solutions are studied. The imETF appro solitons applications in many scientific and engineering fields develops mathematical methodologies, helps research solitons, and advances our knowledge of nonlinear processes. The differences between the results of this investigation and those obtained earlier using alternative methods are analyzed. In terms of fractionality, unconstrained parameters, and applied method sense, all generated wave solutions are found to be new. Physical explanations and a visual representation of the effects of unconstrained parameters and fractionality on the derived solutions are provided. We find that the wave portents change as the number of unconstrained parameters and fractionality increases. In conclusion, we dynamically show that the proper transformation and the applicable imETF method are more effective in analyzing water wave dynamics and might be employed in subsequent studies to shed light on various physical phenomena.

Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

Abstract The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

The construction of exact solution and explicit propagating optical soliton waves of Kuralay equation by the new extended direct algebraic and Nucci’s reduction techniques

ABSTRACT The aim of this paper is to investigate the integrable motion of induced curves using the Kuralay equation, which is a complex integrable coupled system. The soliton solutions derived from Kuralay equation are supposed to represent the most advanced research in several significant phenomena, including optical fibers, nonlinear optics, and ferromagnetic materials. Analytical methods are used to obtain traveling wave solutions for this model as the Cauchy problem cannot be addressed by the inverse scattering transform. In order to find the solitary wave solutions, the new extended direct algebraic and Nucci’s reduction approaches are taken over. As a result, the new extended direct algebraic method provides singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solutions. The Nucci’s reduction technique develops the first integral of differential equation to discuss the conservation and exact solutions. To ensure the sensitivity of the study, the effect of waves on the propagation of solitons and the sensitivity of the model is examined. To illustrate how the fitting values of the system parameters may be utilized to anticipate the behavioral reactions to pulse propagation, the resulting solutions are visually shown in 2D and 3D charts.

M-Fractional soliton solutions of fifth order generalized nonlinear fractional differential equation via (G′/G 2 )-expansion method

The motivation of this research is to investigate four fifth order generalized nonlinear fractional differential equations. Here, we analyze a local derivative, as conformable sense called truncated M-fractional derivative. We also discuss some properties satisfied by the derivative. An effective analytical method namely (G′/G2) -expansion method combined with truncated M-fractional derivative will be studied. As a result, the different types of new soliton solutions were obtained such as dark soliton, bright soliton, kink soliton and periodic soliton wave solutions etc and are verified through MATHEMATICA. This approach is a strong contender for future use in resolving the other problems of a similar nature. A visualization of a few findings is given, which demonstrates the effectiveness and suitability of the suggested approach in producing solutions with well-understood physical behavior. The above mentioned method is more reliable and efficient and simple as compared to rest of the methods to solve the fifth order generalized nonlinear fractional differential equations. The results will also be displayed in two dimensional as well as in three dimensional plots.

Novel waves structures for the nonclassical Sobolev-type equation in unipolar semiconductor with its stability analysis

In this study, the Sobolev-type equation is considered analytically to investigate the solitary wave solutions. The Sobolev-type equations are found in a broad range of fields, such as ecology, fluid dynamics, soil mechanics, and thermodynamics. There are two novel techniques used to explore the solitary wave structures namely as; generalized Riccati equation mapping and modified auxiliary equation (MAE) methods. The different types of abundant families of solutions in the form of dark soliton, bright soliton, solitary wave solutions, mixed singular soliton, mixed dark-bright soliton, periodic wave, and mixed periodic solutions. The linearized stability of the model has been investigated. Solitons behave differently in different circumstances, and their behaviour can be better understood by building unique physical problems with particular boundary conditions (BCs) and starting conditions (ICs) based on accurate soliton solutions. So, the choice of unique physical problems from various solutions is also carried out. The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable. This information can help the researchers in their understanding of the physical conditions.

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

This paper employs a novel $\varphi ^{6}$-model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non-linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton's internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons' three-, two-dimensional, and contour graphs are plotted.

Lump solution and interaction solutions to the fourth-order extended (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper, we explore the exact solutions to the fourth-order extended (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Based on Hirota bilinear method, lump solution, periodic cross-kink solutions and bright-dark soliton solutions were investigated. By calculating and solving, the peak and trough of lump solution are obtained, and the maximum and minimum points of each are solved. The three-dimensional plots and density plots of periodic cross-kink solution and bright-dark soliton solution are drawn and the dynamics of solutions under different parameters are observed.

Optical waves solutions for the perturbed Fokas–Lenells equation through two different methods

This study examines the perturbed Fokas–Lenells equation using two methods: the Bernoulli sub-equation function method and the 1/G′-expansion method. A traveling wave transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The perturbed Fokas–Lenells equation arises in optical fibers, and the article constructs various optical soliton solutions, including periodic dark-bright, singular, and periodic singular soliton solutions. The article also presents the necessary constraint conditions for the existence of valid solitons and graphs 2D, 3D, and contour surfaces based on different parameter values.

Fractional generalized perturbed KdV equation with a power Law kernel: A computational study

In this study, we analyze the nonlinear generalized perturbed KdV equation using the Shehu transform and decomposition approach to obtain solutions. Multiple cases with appropriate initial conditions demonstrate the procedure’s effectiveness and validity, with excellent agreements noted. Simulations reveal three distinct solutions: one bright-soliton, two wave solutions, and dark-bright soliton solutions. Fractional order significantly impacts wave amplitudes and nonlinearity characteristics, affecting system excitations. These findings offer insights into complex behaviors, with potential applications in fluid dynamics, nonlinear optics, and plasma physics, guiding experimental design and system analysis.

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.

Bright-dark soliton solutions and their elastic interaction analysis for a reduced integrable spin Hirota–Maxwell–Bloch equation

Under consideration is a reduced integrable spin Hirota–Maxwell–Bloch (rsHMB) equation, which may have potential applications in describing femtosecond pulse propagation through an erbium doped fiber. With the aid of symbolic computation, we establish the N-fold Darboux transformation (DT) of rsHMB equation based on its known 2 × 2 matrix spectral problem. As an application of the resulting DT, we can get bright-dark soliton and breather solutions of rsHMB equation. Through graphical and asymptotic analysis, the soliton elastic collisions are shown and discussed graphically, and some important soliton physical features including wave velocity, amplitude and energy are analyzed. The consequences and properties presented in this paper may be helpful to comprehend the transmission of optical soliton in erbium doped fiber.

Impacts of a general power law on soliton for a (2+1)-dimensional Zakharov–Kuznetsov equation in magnetized quantum plasmas

In this article, under investigation is a (2+1)-dimensional Zakharov–Kuznetsov equation with a general power law nonlinearity arising from magnetized quantum plasmas model. The analytical periodic wave, bright soliton and bright–dark soliton solutions are computed for the equation via the F-expansion method. Furthermore, the impacts of the general power law parameter on the soliton is examined in details. It is interesting that the general power law can be used to control the amplitude and propagation speed, which can determine partly the kinetic energy. Particularly, as the general power law parameter increases, the soliton’ energy will decrease.

Stochastic Solitons in Birefringent Fibers for Biswas–Arshed Equation with Multiplicative White Noise via Itô Calculus by Modified Extended Mapping Method

Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. This model contains many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. The constraints on the parameters are considered to guarantee the existence of these stochastic solutions. Furthermore, some of the selected solutions are described graphically to demonstrate the physical nature of the obtained solutions.

Abundant solitary wave solutions of Gardner's equation using three effective integration techniques

<abstract><p>Gardner's equation has been discussed in the article for finding new solitary wave solutions. Three efficient integration techniques, namely, the Kudryashov's <italic>R</italic> function method, the generalized projective Ricatti method and $ \frac{G'}{G^2} $-expansion method are implemented to obtain new dark soliton, bright soliton, singular soliton, and combo soliton solutions. Moreover, some of the obtained solutions are graphically depicted by using $ 3 $D-surface plots and the corresponding $ 2 $D-contour graphs.</p></abstract>

Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

<abstract> <p>This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.</p> </abstract>

EXAMINATION OF KRAENKEL-MANNA-MERLE SYSTEM BY SINE-GORDON EXPANSION METHOD

In this study, Kraenkel-Manna-Merle (KMM) system is discussed. Sine-Gordon expansion method (SGEM), which is one of the solution methods of nonlinear evolution equations (NLEEs), has been applied to this system. Thus, by applying this method for the first time, some dark soliton, bright soliton, and dark-bright soliton solutions of the KMM system have been obtained. In addition, by giving specific values to the achieved solutions, 2D and 3D graphics of the solutions were plotted by way of the Wolfram Mathematica 12 program.

Optical solitons in Fiber Bragg Gratings for the coupled form of the nonlinear [formula omitted]-dimensional Kundu-Mukherjee-Naskar equation via four powerful techniques

In this article, four powerful techniques namely, the three wave method, double exponential, homoclinic breather, M-Shaped rational, M-Shaped with one kink, and M-Shaped with two kink are putting into practice to bolster the new optical solutions of the coupled form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation in Fiber Bragg Gratings (FBGs). Implementation of the appropriate functions of the solutions results in a different form of multi-wave solutions with their interaction phenomena such as dark soliton solutions, singular soliton solutions, bright soliton solutions, kink soliton solutions, Multi-Peak soliton solutions, multi dark-bright soliton solutions, multi bright soliton solutions, multi dark soliton solutions with different structure, multi M-Shaped soliton solutions, multi M-Shaped soliton solutions, M-Shaped soliton solutions with different structure, kink soliton solutions with different structure, and periodical M-Shaped soliton solutions. Three-dimensional graphics demonstrating the dependability and productivity of the suggested methodologies are used to describe the dynamics of the produced solutions.

Pattern Formation of a Bubbly Fluid Mixture under the Effect of Thermodynamics via Kudryashov-Sinelshchikov Model

In this paper, new explicit wave solutions via liquid-gas bubbles are obtained for the fractional Kudryashov-Sinelshchikov (KS) equation under thermodynamic assumptions. A new fractional definition is applied to get these solutions that are utilized to represent the phenomenon of pressure waves under thermodynamic conditions. Two analytical techniques are used to explore the model which is sinh-Gorden equation expansion and Riccati-Bernoulli Sub-ODE methods. These approaches provide complex hyperbolic, hyperbolic, complex trigonometric, and trigonometric solutions for the fractional KS equation, particularly singular, combined singular, dark, bright, combined dark-bright, and other soliton solutions. Furthermore, acquired results are illustrated by 3D graphs for suitable parametric values that highlight the physical importance and dynamical behaviors of the equation. It is also demonstrated that the purposed approaches are powerful strategies for developing exact traveling wave solutions for a wide range of problems found in mathematical sciences.