Exact Traveling Wave Solutions Research Articles

The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method

The purpose of this article is to explore different types of solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which is termed as KP-Gardner equation, extensively used to model strong nonlinear internal waves in ( 1 + 2 )-dimensions on the stratified ocean shelf. This evolution equation is also used to describe weakly nonlinear shallow-water wave and dispersive interracial waves traveling in a mildly rotating channel with slowly varying topography. Introducing Liu’s approach regarding the complete discrimination system for polynomial and the trial equation technique, a set of new solutions to the KP-mKP equation containing Jacobi elliptic function have been derived. It is found that these analytical solutions numerically exhibit different nonlinear structures such as solitary waves, shock waves, and periodic wave profiles. The reliability and effectiveness are confirmed from the numerical graphs of the solutions. Finally, the existence and validity of the various topological structures of the solutions are confirmed from the phase portrait of the dynamical system. Based on this investigation, it is confirmed that the method is not only suited for obtaining the classification of the solutions but also for qualitative analysis, which means that it can also be extended to other fields of application.

ABUNDANT EXACT TRAVELING WAVE SOLUTIONS TO THE LOCAL FRACTIONAL (3+1)-DIMENSIONAL BOITI–LEON–MANNA–PEMPINELLI EQUATION

Based on the local fractional derivative, we develop a new local fractional (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. With the help of the traveling wave transform of the nondifferentiable type, we convert the local fractional equation into a nonlinear local fractional ordinary differential equation (ODE). Then the Mittag-Leffler function-based method is presented here for the first time to construct the abundant nondifferentiable exact traveling wave solutions, and three families (six sets) of the exact solutions are obtained. In addition, the performances of different solutions on the Cantor set are presented via numerical results in the form of 3D plots. It reveals that the proposed method is simple and straightforward, which can be used to study the local fractional partial differential equations (PDEs).

A unique computational investigation of the exact traveling wave solutions for the fractional-order Kaup-Boussinesq and generalized Hirota Satsuma coupled KdV systems arising from water waves and interaction of long waves

A novel technique, named auxiliary equation method, is applied in this research work for obtaining new traveling wave solutions for two interesting proposed systems: the Kaup-Boussinesq system and generalized Hirota-Satsuma coupled KdV system with beta time fractional derivative. Our solutions were obtained using MAPLE software. This technique shows a great potential to be applied in solving various nonlinear fractional differential equations arising from mathematical physics and ocean engineering. Since a standard equation has not been used as an auxiliary equation for this technique, different and novel solutions are obtained via this technique.

A GENERALIZED METHOD AND ITS APPLICATIONS TO n-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS IN FRACTAL DOMAIN

A generalized local fractional Riccati differential equation (LFRDE) method, which is a combination of the local fractional Riccati differential equation method and the transformed rational function approach, is presented to obtain the non-differentiable exact traveling wave solutions for n-dimensional fractional partial differential equations. In order to verify the effectiveness of the method, the seventh-order local fractional Sawada–Kotera–Ito equation, the local fractional sine-Gordon equation and the local fractional Kadomtsev–Petviashvili equation in fractal domain are first considered. The obtained results show that the presented method involving fractal special functions, are powerful and effective for obtaining exact solutions of nonlinear fractional partial differential equations in fractal domains.

Fractional derivative-based performance analysis to Caudrey–Dodd–Gibbon–Sawada–Kotera equation

This paper explores the innovative soliton solutions to the fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation with Beta and Atangana–Baleanu (AB) fractional derivatives. The new auxiliary equation method (NAEM) is used to extricate novel analytical solutions to the CDGSK equation. A comparative analysis of the solutions is conducted by implementing these fractional derivatives. The impact of different values of fractional parameter on the behavior of obtained solutions is also explored. The shapes and dynamics of the retrieved solutions are portrayed employing 3D and 2D graphs. These new solitary wave solutions are shown to encompass dark, bright, periodic, and singular wave solutions. Analytical solutions expound that this method is an effective mathematical tool to find exact traveling wave solutions to nonlinear models found in several fields of science and engineering.

Exact traveling wave solutions for the system of the ion sound and Langmuir waves by using three effective methods

This paper gives a study to the system ofion sound andLangmuir waves (SEISLWs). Seven sets of the traveling wave solutions (TWSs) like the bright solitary (Type-1, Type-2, Type-3, Type-4), kinky bright solitary, dark solitary and periodic wave solutions are developed through three effective methods, which are simplified extended tanh-function method, He’s frequency formulation and the energy balance theory. The dynamic behaviors of the absolute, real and imaginary parts of the solutions are also plotted via the 3-D contours. The results in this work fully prove that the proposed methods are powerful tools to construct abundant TWS for the PDEs arising in physics.

Exact solutions of the different dimensional CBS equations in mathematical physics

The focus of this article is to find the exact traveling wave solutions to Calogero–Bogoyavlenskii–Schiff (CBS) equation by employing the exp−Φζ expansion method. The traveling wave solutions are expressed by the hyperbolic function solutions, trigonometric function solutions and rational function solutions. 3D and 2D charts are drawn from the obtained solutions by Mathematica software and discussed the graphical behavior of solutions. Finally, an effective mathematical tools for solving nonlinear evolution equations in mathematical physics are provided by the proposed method. This method is more common and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations.

Some new type optical and the other soliton solutions of coupled nonlinear Hirota equation

In this study, we obtained some exact traveling wave solutions of coupled nonlinear Hirota equation (CNHE) by using generalized G'G-expansion method. The solutions we obtained are hyperbolic, trigonometric and exponential solutions. Analytical results for the obtained solutions for the CNHE have been corroborated through the Mathematica 11.2 symbolic calculation. Solutions have been illustrated in 3D graphs, and 2D plots. The present method has been used to obtain exact traveling wave solutions for several nonlinear partial differential equations.

Analytical and Data-Driven Wave Approximations of an Extended Schrödinger Equation

Using both analytical and numerical techniques, we discuss wave solutions within the framework of an extended nonlinear Schrödinger equation with constant coefficients equipped with spatiotemporal dispersion, self-steepening effects, and a Raman scattering term. We present the exact traveling wave solution of the system in terms of Jacobi elliptic functions and mention some symmetry results as they relate to the resulting ordinary differential equation. A constructed bright soliton solution serves as the base to compare a numerical solution of the system using spectral Fourier methods with a precise statistical low-rank approximation using a data-driven approach aided by the Koopman operator theory. We found that the spatiotemporal feature added to the model serves as a regularizing tool that enables a precise reconstruction of the original solution.

Topological properties and wave structures of Gilson–Pickering equation

The aim of this paper is to present the topological properties and wave structures of the Gilson–Pickering equation. By the traveling wave transformation, the corresponding traveling wave system of the original equation is obtained, and then a conserved quantity, namely the Hamiltonian is constructed via it. After that, the existences of the soliton and periodic solutions are established by the bifurcation method. To verify our conclusion explicitly, the corresponding exact traveling wave solutions are constructed. In particular, via the generalized trial equation method which is proposed in this paper, a special kind of soliton solution, namely the Gaussian soliton solution is given. To the best of our knowledge, this is the first time that a Gaussian soliton solution has been constructed to an equation with no logarithmic nonlinearity.

Computational wave solutions of some nonlinear evolution equations

This paper aims to realize some of the vital nonlinear evolution of partial differential equations in mathematical physics. We implement an extended simple equation (ESE) method to get exact and solitary traveling wave solutions of the Calogero-Degasperis (CD) equation, potential Kadomtsev–Petviashvili (pKP) equation, generalized Korteweg–de Vries (gKdV) equation, and the (2 + 1)–dimensional Zoomeron (Z) equation. We obtain a new form of solutions for each model as mentioned above. This method is considered a new, robust, practical, and direct method. As we see, the remarkable ability of this method to apply on the fractional and integer order of nonlinear partial differential equations.

Travelling wave solutions for the doubly dispersive equation using improved modified extended tanh-function method

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation.

The classification of the single traveling wave solutions to the KdV equation with higher-order nonlinearity

The Korteweg–de Vries (KdV) equation with higher-order nonlinearity is considered as the model of the wave propagation in one-dimensional nonlinear lattice. In this paper, we obtain the exact solutions to the KdV equation with higher-order nonlinearity. The complete discrimination system for polynomial method is applied to handle such a problem. All the possible exact traveling wave solutions are obtained.

Bifurcation Analysis and Single Traveling Wave Solutions of the Variable-Coefficient Davey–Stewartson System

This paper mainly studies the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system. By employing the traveling wave transformation, the variable-coefficient Davey–Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations. On the one hand, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. On the other hand, we use the polynomial complete discriminant method to obtain the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.

Exact traveling wave solutions of the Schamel Burgers’ equation by using generalized-improved and generalized [formula omitted] expansion methods

The Schamel Burgers’ equation is a nonlinear partial differential equation which produces the shock type traveling waves in extraordinary physical situations. Previously, this sort of equation is solved with well known tan−hyperbolic method. In this paper, we present the exact solutions of the Schamel Burgers’ equation using generalized-improved G′G and generalized G′G expansion methods with the assistance of Mathematica-12 and observe the behavior of the solutions both analytically and numerically with requisite graphs. The solutions of mentioned equation admits the shock like dynamical structures.

On the study of solutions of Bogoyavlenskii equation via improved $ G'/G^2 $ method and simplified $ \tan(\phi(\xi)/2) $ method

<abstract><p>The Bogoyavlenskii equation is used to describe some kinds of waves on the sea surface and discussed by many researchers. Recently, the $ G'/G^2 $ method and simplified $ \tan(\frac{\phi(\xi)}{2}) $ method are introduced to find novel solutions to differential equations. To the best of our knowledge, the Bogoyavlenskii equation has not been investigated by these two methods. In this article, we applied these two methods to the Bogoyavlenskii equation in order to obtain the novel exact traveling wave solutions. Consequently, we found that some new rational functions, trigonometric functions, and hyperbolic functions can be the traveling wave solutions of this equation. Some of these solutions we obtained have not been reported in the former literature. Through comparison, we see that the two methods are more effective than the previous methods for this equation. In order to make these solutions more obvious, we draw some 3D and 2D plots of them.</p></abstract>

Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures liquid and gas bubbles

In this paper, some new exact traveling and oscillatory wave solutions to the Kudryashov-Sinelshchikov non-linear PDE are investigated by using Bernoulli sub-equation function method. Profiles of obtained solutions are plotted.

The Improved Modified Extended Tanh-Function Method to Develop the Exact Travelling Wave Solutions of a Family of 3d Fractional Wbbm Equations

The Improved Modified Extended Tanh-Function Method to Develop the Exact Travelling Wave Solutions of a Family of 3d Fractional Wbbm Equations

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.

New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.