Abundant Solitary Wave Solutions Research Articles

Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative

In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, we generalize some previous results. As the Boiti–Leon–Manna–Pempinelli equation is a model for an incompressible fluid, the solutions obtained may be utilized to represent a wide variety of fascinating physical phenomena. We construct a large number of 2D and 3D figures to demonstrate the impact of the M-truncated derivative on the exact solution of the BLMPE-MTD.

Abundant solitary wave solutions of the higher dimensional generalized Camassa–Holm–KP model in shallow water waves

In this study, we construct the abundant families of soliton solutions for the generalized (2+1)-dimensional Camassa–Holm–KP equation. This equation is used in tiny amplitude shallow water waves. The solutions are obtained in the form of Jacobi elliptic function solutions which are the soliton and solitary wave solutions. They are observed in the form of shock, bright, combined complex shock, singular and periodic waves, as well as rational solutions. Some techniques are applied to obtain the solutions with the help of Mathematica namely as; ϕ6-model expansion and a new modified extended direct algebraic method. In nonlinear dynamics, the acquired results are particularly helpful for studying and confirming the analytical solutions with numerical and experimental work. Additionally, the contour diagrams and three- and two-dimensional representations of the solutions are drawn for the various parameter values.

Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications

The unstable nonlinear Schrödinger models illustrate the time evolution of disturbances in marginally stable or unstable media. In this article, an improved auxiliary equation method is proposed for conformable fractional-order nonlinear evolution equations to obtain the novel wave solutions of space–time fractional unstable and modified unstable nonlinear Schrödinger equations. Several types of novel wave solutions are constructed in forms of some special Jacobi elliptic functions, rational, exponential, trigonometric and hyperbolic functions in which various are unique and having key utilization in quantum physics. With the aid of appropriate values to parameters, different shapes of bright, dark, kink type soliton, multi-peak solitons, periodic solitary waves etc. are depicted. These distinct physical structures are helpful in the understanding the complex physical interpretation of unstable dynamical models. Furthermore, this article gives an idea, how can reduce the conformable fractional order unstable nonlinear Schrödinger equations into an ODE of one variable to obtain the exact solutions. The numerous obtained waves and other solutions reveal the usefulness of the proposed technique which can be utilized to other nonlinear fractional models arising in quantum physics and other fields of applied sciences.

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>

Bifurcation and traveling wave solutions of stochastic Manakov model with multiplicative white noise in birefringent fibers

Manakov model is an extended form of Schrödinger equation. It is a coupled partial differential equation governing the dual-mode transmission of optical fiber communication and the deep-sea transportation with double-layer constraints. In this paper, the bifurcation and traveling wave solutions of the stochastic Manakov model(SMM) describing the multiplicative white noise of optical signal propagation in birefringent fibers are studied. The research work is carried out in the following three steps. Firstly, by the help of traveling wave transformation and first integral, the SMM with multiplicative white noise is simplified to a 2D planar dynamic system. Secondly, the phase diagrams of the planar dynamic system are drawn to determine the possible traveling wave solutions. Finally, based on the classification of parameter group, combined with the elliptic integral technique, the solitary wave solutions and traveling wave solutions of SMM are constructed. The obtained bifurcation results reveal the dynamic behavior of SMM from a geometric point of view. According to the phase portraits, we get abundant solitary wave solutions and traveling wave solutions, including elliptic function periodic solution, trigonometric function solution and singular periodic solution. This is the first study on the bifurcation of SMM with multiplicative white noise, and the obtained results provide the propagation of optical solitons in nonlinear optics.

Construction of traveling wave solutions of the (2 + 1)‐dimensional modified KdV‐KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)‐dimensional modified KdV‐KP equation are derived by employing an ansatz method, named the enhanced (G′/G)‐expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum‐modified KdV‐KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum‐modified KdV‐KP system supports solitary waves having different shapes and speeds for different values of the parameters.

RETRACTED: Abundant solitary wave solutions to a perturbed Schrödinger equation with Kerr law nonlinearity via a novel approach

The main purpose of the present paper is to introduce a reliable method, for the first time, in solving differential equations with partial derivatives. The significant idea behind this method is the modification of a well-known method. The main base functions used in the solution’s structures are Jacobi elliptic functions which are known functions and have many applications in practice. In order to achieve these results, the perturbed Schrödinger equation with Kerr law nonlinearity is considered. This nonlinear equation illustrates the propagation of optical solitons in nonlinear optical fibers. Thereupon, several analytical solutions corresponding to this model are obtained through employing an improved generalized exponential rational function method. Then we present a new version of it that is presented for the first time in the literature. Using these methods, several wave solutions related to the considered model are obtained. Moreover, several numerical simulations relevant to the resulting solutions are performed in the paper. As a notable features of the proposed method, the obtained solutions are characterized by the Jacobi elliptic functions. These solutions contain an index that for some specific choices is reduced to standard functions such as trigonometric and hyperbolic. The obtained results and solutions can be used for investigating the mechanism of several nonlinear phenomena laboratory and space plasmas. All necessary calculations are performed via symbolic packages of Wolfram Mathematica.

Abundant solitary wave solutions of the Chen–Lee–Liu equation via a novel analytical technique

Abundant solitary wave solutions of the Chen–Lee–Liu equation via a novel analytical technique

New Solitary Wave Solutions of the Fisher Equation

In this paper, we use Riccati equation to find new solitary wave solutions of Fisher equation, which describes the process of interaction between diffusion and reaction. It is of great importance to comprehend the equation to solve the problems in chemical kinetics and population dynamics. We resolve the Ricatti equation through diverse function transformation and many types of exact solutions are obtained. Then it is used as an auxiliary equation to solve Fisher equation. In the process, we select different coefficients in the Racatti equation, as a result, abundant solitary wave solutions are obtained, some of which haven’t been found in other documents yet. Moreover, these solutions we got in this paper will be favorable for understanding the Fisher equation.

Soliton solutions for fractional DNA Peyrard-Bishop equation via the extended -expansion method

In this article, the dynamical DNA equation comes into existence in the oscillatory chain, categorized as the Peyrard-Bishop model. The fractional Peyrard-Bishop model has been studied via β-derivative and M-truncated derivative, for obtaining abundant solitary wave solutions. The proposed model is examined via M-truncated derivative for the first time in this article. The traveling wave transformation method is used to convert the proposed model into an ordinary differential equation. The extended -expansion method is applied to extract dark solitons, periodic solutions, singular solitons, combo solitons and rational solutions. On comparing our extracted solutions with the existing solutions in the literature, it is found that our results are new and not found in the literature. The obtained results are also explained graphically by plotting 3D plots to understand the phenomena of the proposed model.

Abundant solitary wave solutions for the fractional coupled Jaulent–Miodek equations arising in applied physics

This article explores the abundant solitary wave solutions of the conformable coupled Jaulent–Miodek (JM) equations appearing in applied physics. The aforesaid coupled equations belong to the family of shallow-water wave equations. Two recent modified integration schemes are used for the first time to produce a novel solitary wave, trigonometric and other solutions with some free parameters in the conformable derivative sense. In particular, the modified Kudryashov and [Formula: see text]-expansion schemes are used to illustrate the wave propagations through aforesaid solutions of the JM equations. Furthermore, a comparison is made with some recent results and the dynamics of the obtained solutions are displayed for the reader via soft computation. The outcomes reveal that the methods are effective and provide a direct way of finding novel solutions.

Soliton solutions to the DNA Peyrard–Bishop equation with beta-derivative via three distinctive approaches

In this paper, we explore the DNA dynamic equation arising in the oscillator-chain named as Peyrard–Bishop model for abundant solitary wave solutions. The aforesaid model is studied for the first time via a novel fractional derivative operator. A modified traveling wave transformation with the aforementioned operator is used to transform the space-time fractional Peyrard–Bishop model into an ODE. Then, the three prolific schemes namely the simplest equation method, the Kudryashov method and the modified $$(G'/G^2)$$ -expansion method are employed. These methods contribute a variety of exact soliton solutions including the bright, dark and singular solution. Also the obtained solutions are verified through symbolic soft computations. Furthermore, some results are also explained through numerical simulations that show the novelty of our work as compared to the existing literature about the classical Peyrard–Bishop model.

Abundant solitary wave solutions to an extended nonlinear Schr\xf6dinger\u2019s equation with conformable derivative using an efficient integration method

The prevalence of the use of mathematical software has dramatically influenced the evolution of differential equations. The use of these useful tools leads to faster advances in the presentation of numerical and analytical methods. This paper retrieves several soliton solutions to the fractional perturbed Schrödinger’s equation with Kerr and parabolic law nonlinearity, and local conformable derivative. The method used in this article, called the generalized exponential rational function method, also relies heavily on the use of symbolic software such as Maple. The considered model has prominent applications in water optical metamaterials. The method retrieves several exponential, hyperbolic, and trigonometric function solutions to the model. The numerical evolution of the obtained solutions is also exhibited. The resulted wide range of solutions derived from the method proves its effectiveness in solving the model under investigation. It is also recommended to use the technique used in this article to solve similar problems.

Tanh–coth scheme for traveling wave solutions for Nonlinear Wave Interaction model

Abstract The coupled 1D nonlinear Schrodinger Zakharov System (sch-zakh) is considered as the model equation for Nonlinear Wave Interactions model. Tanh–coth Scheme is used to derive abundant solitary wave solutions for the model equations. The obtained solutions include soliton solutions.

New extension of the tanh-function method and application to the Whitham–Broer–Kaup shallow water model with symbolic computation

Abstract In this Letter, the tanh-function method is further extended by introducing a more generalized expansion form. Compared with other existing tanh-function methods, our algorithm provides the possibility for constructing more types of travelling-wave solutions (especially the solitary-wave solutions) for a given system of nonlinear evolution equations (NLEEs). Applying such an algorithm to the Whitham–Broer–Kaup shallow water model with symbolic computation, abundant solitary-wave solutions are obtained and some novel solitary-wave structures are also revealed under certain parametric conditions. After suitable improvements, this algorithm can also be applicable to the construction of nontravelling solitary-wave solutions for variable-coefficient and higher-dimensional NLEEs.

The extended tanh method for abundant solitary wave solutions of nonlinear wave equations

The extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations. The obtained solutions include solitons, kinks and plane periodic solutions. The study is an extension to the remarkable development by Malfliet [1]. The extended tanh method presents a wider applicability for handling nonlinear wave equations.

Exact Solutions of the Coupled KdV System via a Formally Variable Separation Approach**The project supported by National Natural Science Foundation and “Scaling Plan” of China, and Natural Science Foundation of Zhejiang Province of China

Most of the nonlinear physics systems are essentially nonintegrable. There is no very good analytical approach to solve nonintegrable system. The variable separation approach is a powerful method in linear physics. In this letter, the formal variable separation approach is established to solve the generalized nonlinear mathematical physics equation. The method is valid not only for integrable models but also for nonintegrable models. Taking a nonintegrable coupled KdV equation system as a simple example, abundant solitary wave solutions and conoid wave solutions are revealed.