Dodd Bullough Mikhailov Research Articles

Attached Flows for Reaction–Diffusion Processes Described by a Generalized Dodd–Bullough–Mikhailov Equation

This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction–diffusion equations, the generalized Dodd–Bulough–Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation.

New traveling‐wave solutions for a nonlinear Dodd–Bullough–Mikhailov equation

In this article, we will present new and more general traveling‐wave solutions for the Dodd–Bullough–Mikhailov equation based on the extended Fan sub‐equation method. The two‐dimensional affine sphere's intrinsic geometry is determined by Dodd–Bullough in the three‐dimensional unimodular affine space. The most significant accomplishment of our method is that we are able to provide all forms of the previously obtained solutions in one action, using at least four different approaches. As a result, the general elliptic equation with five parameters can be connected to other sub‐equations with three parameters that already exist, such as the Riccati equation, the first sort of elliptic equation, the auxiliary ordinary equation, and the modified Riccati equation. The extended Fan sub‐equation method is utilized to construct traveling‐wave solutions, solitary wave solutions, dark soliton solutions, and bell soliton solutions for the nonlinear Dodd–Bullough–Mikhailov equation. Mathematica is utilized to depict the dynamics of various wave structures in 3D, 2D, and contour formats, utilizing a specific set of parameters. Our graphical comparative analysis suggests that the employed method is both reliable and powerful for obtaining exact solutions of nonlinear evolution equations. The approach employed in this study may also be applicable in enhancing the understanding of numerous other complex physical phenomena.

Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations

This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case.

Exploring exact solutions for physical differential models through generalized derivatives

In this study, we develop three well-known fractional differential physical models with novel exact solutions. Liouville, Dodd-Bullough-Mikhailov (DBM), and Sinh-Gordon equations are the models under consideration. These models will be broken down into three nonlinear ordinary differential equations using a waveform transformation, which can be precisely solved using the approach of the simplest equation method. The suggested method is applicable to several categories of nonlinear physical models and allows us to extract numerous generalized solutions in soliton and periodic forms The resulting solutions may also be directly compared with a number of findings obtained in the literature. Additionally, representations in two and three dimensions are provided to show how changing the fractional parameter’s amount may impact how monotonic the solutions are obtained.

The Functional Expansion Approach for Solving NPDEs as a Generalization of the Kudryashov and G′/G Methods

This paper presents the functional expansion approach as a generalized method for finding traveling wave solutions of various nonlinear partial differential equations. The approach can be seen as a combination of the Kudryashov and G′/G solving methods. It allowed the extension of the first method to the use of second order auxiliary equations, and, at the same time, it allowed non-standard G′/G-solutions to be generated. The functional expansion is illustrated here on the Dodd–Bullough–Mikhailov model, using a linear second order ordinary differential equation as an auxiliary equation.

Exact solutions of conformable fractional differential equations

This article is about to formulate exact solutions of the time fractional Dodd-Bullough-Mikhailov (DBM) equation, Sinh-Gordon equation and Liouville equation by utilizing simplest equation method (SEM) in conformable fractional derivative (CFD) sense. Using a proper transformation, the original equations are transformed to nonlinear ordinary differential equations (ODEs). The method is very simple in comparison with the classical techniques and very much effective for solving fractional order partial differential equations (FOPDEs).

The simplest equation approach for solving non-linear Tzitzéica type equations in non-linear optics

The aim of this work is to investigate new exact solutions of Tzitzéica type equations. We utilize the Painlevé transformation to transform the aforesaid non-linear evolution equations into ordinary differential equations. Then, the simplest equation method is employed for securing some real and complex solutions of the Tzitzéica equation, the Tzitzéica–Dodd–Bullough equation and the Dodd–Bullough–Mikhailov equation. After the execution of the simplest equation method, we obtain many new results more simply and reliably than the other approaches executed on these equations. The solutions are obtained and verified through soft computations. Also, the dynamics of some solutions are presented via three types of graphs including 2D, 3D and contour graphs.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon equation

This paper concerns the qualitative behavior and exact traveling wave solutions of the generalized double combined sinh–cosh-Gordon equation. This equation generalizes sinh-Gordon equation, Liouville equation, Dodd–Bullough–Mikhailov equation, Tzitzeica–Dodd-Bullough equation and Zhiber-Shabat equation as special cases. Thus, this paper presents a unified analysis to find the exact solutions of these known equations. Our results generalize many previous known results (Chen et al., 2009; Fan et al., 2011; Geng et al., 2007; He and Meng, 2017; He et al. 2014; Li and Li 2005; Seadawy et al. 2017; Tang and Huang 2007). We find different kinds of exact solutions such as bright soliton, dark soliton, kink wave, anti-kink wave solutions and periodic wave solutions. Moreover, the explicit expressions of the bounded exact traveling wave solutions are given. Finally, a conclusion ends this paper.

Structure of traveling wave solutions for some nonlinear models via modified mathematical method

Abstract We have employed the exp(-φ(ξ))-expansion method to derive traveling waves solutions of breaking solition (BS), Zakharov-Kuznetsov-Burgers (ZKB), Ablowitz-Kaup-Newell-Segur (AKNS) water wave, Unstable nonlinear Schrödinger (UNLS) and Dodd-Bullough-Mikhailov (DBM) equations. These models have valuable applications in mathematical physics. The results of the constructed model, along with some graphical representations provide the basic knowlegde about these models. The derived results have various applications in applied science.

The solitons and periodic travelling wave solutions for Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations in quantum field theory

In this paper, new types of Jacobi elliptic function solutions of Dodd–Bullough–Mikhailov(DBM) and Tzitzeica–Dodd–Bullough(TDB) equations have been obtained using a new extended auxiliary equation method. A new family of explicit travelling wave solutions is derived. The solitary wave solutions and periodic solutions for these equations are formally derived from the Jacobi elliptic function solutions. The proposed method has been efficiently applied to solve the DBM and TDB equations.

New optical solitons of Tzitzeíca type evolution equations using extended trial approach

The properties of Tzitzeica equations in nonlinear optics have been the discussion of many recent studies. In this article, a new and productive implementation of trial equation method is chosen to handle this class of nonlinear evolution equations. As a result, the new and more exact periodic, singular, hyperbolic and rational solutions of Tzitzeica, Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations are formally obtained. The extended trial equation method along with the symbolic computation gives a versatile technique to deal such kinds of nonlinear evolution equations in nonlinear optics.

Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd–Bullough–Mikhailov equation

In this study, the modified exp $$ ( - \Phi (\eta )) $$ -expansion function method is used in constructing some solitary wave solutions to the Oskolkov–Benjamin–Bona–Mahony–Burgers, one-dimensional Oskolkov equations and the Dodd–Bullough–Mikhailov equation. We successfully construct some singular solitons and singular periodic waves solutions with the hyperbolic, trigonometric and exponential function structures to these three nonlinear models. Under the choice of some suitable values of the parameters involved, we plot the 2D and 3D graphics to some of the obtained solutions in this study. All the obtained solutions in this study verify their corresponding equation. We perform all the computations in this study with the help of the Wolfram Mathematica software. The obtained solutions in this study may be helpful in explaining some practical physical problems.

New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov and Tzitzéica–Dodd–Bullough equations, are obtained using the expa function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.

The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics

In this paper, using the Painleve property, the traveling wave transformation, and the sine-Gordon expansion method (SGEM), the Tzitzéica type evolution equations in nonlinear optics, including the Tzitzéica equation, the Dodd–Bullough–Mikhailov (DBM) equation, the Tzitzéica–Dodd–Bullough (TDB) equation, and the Liouville equation are studied. As a result, a series of traveling wave solutions for the Tzitzéica type equations are obtained. The efficiency of the solution methodology in exactly solving nonlinear evolution equations is confirmed.

New exact traveling wave solutions of the Tzitzéica type equations using a novel exponential rational function method

The Tzitzéica type equations arising in nonlinear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov, and Tzitzéica–Dodd–Bullough equations are solved analytically, in the present paper. A solution technique in the area of computational mathematics called the novel exponential rational function method is formally proposed for this purpose. As a consequence, a series of new exact solutions for the Tzitzéica type equations are obtained, proving the super performance of novel exponential rational function method.

New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved $$\tan \left( {\varPhi \left( \xi \right)/2} \right)$$ tan Φ ξ / 2 -expansion method

The paper deals with the Tzitzeica type nonlinear evolution equations arising in nonlinear optics and their new exact solutions. First, through the use of the Painleve transformation and Lie symmetry method, the Tzitzeica, Dodd–Bullough–Mikhailov, and Tzitzeica–Dodd–Bullough equations are converted to nonlinear ordinary differential equations (NODEs), and then, a modified version of the improved $$\tan \left( {\varPhi \left( \xi \right)/2} \right)$$ -expansion method, proposed by the authors, is adopted to generate new exact solutions of the reduced equations. The method truly recommends a reliable and capable technique to produce new exact solutions of a variety of nonlinear partial differential equations (NPDEs).

Bifurcations of traveling wave solutions for Dodd–Bullough–Mikhailov equation and coupled Higgs equation and their applications

In this article, we shall introduce a new and more general traveling wave solutions for Dodd–Bullough–Mikhailov equation and coupled Higgs equation according to the modified extended tanh-function method with the aid of Maple 16. The important fact of this method is to explain the solitary wave solutions for nonlinear partial differential equations (NLPDEs.) which illustrate the physical phenomena and help other researchers for investigating the stability of Dodd–Bullough–Mikhailov equation and coupled Higgs equation. The traveling wave solutions; solitary wave solutions; dark and bell soliton solutions of nonlinear Dodd–Bullough–Mikhailov and coupled Higgs dynamical equations are constructed by employing modified extended tanh method, which have important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the traveling wave solutions; solitary wave solutions; dark and bell soliton solutions for these equations. Comparison between our results and the well-known results will be presented.

Meshless numerical analysis of a class of nonlinear generalized Klein–Gordon equations with a well-posed moving least squares approximation

• A meshless method is developed for some generalized Klein–Gordon equations. • A convergent iterative scheme is performed to tackle the nonlinearity. • Examples involving five kinds of nonlinear equations and solitons are given. • Numerical results indicate that the method is about second-order convergence in both time and space. This paper presents a meshless method for the numerical solution of a class of nonlinear generalized Klein–Gordon equations. In this method, a time discrete technique is first adopted to discretize the time derivatives, and then a well-posed moving least squares (WP-MLS) approximation using shifted and scaled orthogonal basis functions is developed to approximate the spatial derivatives. To deal with the nonlinearity, an iterative scheme is presented and the corresponding convergence is discussed theoretically. Numerical examples involving Klein–Gordon, Dodd–Bullough–Mikhailov, sine-Gordon, double sine-Gordon and sinh-Gordon equations, and line and ring solitons are provided to illustrate the performance and efficiency of the method.

New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics

The properties of Tzitzéica equations in non-linear optics have been the subject of many recent studies. In this article, a new and effective modification of Kudryashov method is adopted to study this class of non-linear evolution equations. As an achievement, new exact traveling wave solutions of Tzitzéica, Dodd–Bullough–Mikhailov (DBM) and Tzitzéica–Dodd–Bullough (TDB) equations are formally extracted. It is believed that the modified Kudryashov method along with the symbolic computation package suggests a promising technique to handle non-linear evolution equations in non-linear optics.