Mikhailov Equation 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.

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.

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.

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.

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.

Exact travelling wave solutions of some nonlinear equations by [formula omitted]-expansion method

Here we find exact solutions of some nonlinear evolution equations within the framework of the G ′ G -expansion method. Exact solutions of five nonlinear equations of physical importance viz. the coupled Schrodinger–KdV equation, the coupled nonlinear Reaction–Diffusion equation, the Foam Drainage equation, the Phi-Four equation and the Dodd–Bullough–Mikhailov equation are obtained. These general solutions can be reduced in some standard results derived by some other methods. Two and three dimensional plots of some of the results are also presented.

Tzitzeica solitons versus relativistic Calogero–Moser three-body clusters

We establish a connection between the hyperbolic relativistic Calogero–Moser systems and a class of soliton solutions to the Tzitzeica equation (also called the Dodd–Bullough–Zhiber–Shabat–Mikhailov equation). In the 6N-dimensional phase space Ω of the relativistic systems with 2N particles and N antiparticles, there exists a 2N-dimensional Poincaré-invariant submanifold ΩP corresponding to N free particles and N bound particle-antiparticle pairs in their ground state. The Tzitzeica N-soliton tau functions under consideration are real valued and obtained via the dual Lax matrix evaluated in points of ΩP. This correspondence leads to a picture of the soliton as a cluster of two particles and one antiparticle in their lowest internal energy state.

The rational dressing for abelian twisted loop Toda systems

We consider abelian twisted loop Toda equations associated with the complex general linear groups. The Dodd--Bullough--Mikhailov equation is a simplest particular case of the equations under consideration. We construct new soliton solutions of these Toda equations using the method of rational dressing.

Exp-function method for nonlinear wave equations

Abstract In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Generalized tanh Method Extended to Special Types of Nonlinear Equations

By some ‘pre-possessing’ techniques we extend the generalized tanh method to special types of nonlinear equations for constructing their multiple travelling wave solutions. The efficiency of the method can be demonstrated for a large variety of special equations such as those considered in this paper, double sine-Gordon equation, (2+1)-dimensional sine-Gordon equation, Dodd-Bullough- Mikhailov equation, coupled Schrödinger-KdV equation and (2+1)-dimensional coupled Davey- Stewartson equation. - Pacs: 03.40.Kf; 02.30.Jr.

Applications of extended tanh method to ‘special’ types of nonlinear equations

In this paper, we explore more applications of extended tanh method to some special nonlinear equations. Such equations are not of differential polynomial form so that cannot be directly dealt with by tanh method or extended tanh method. As examples, we apply a recently proposed extended tanh method to build some new explicit solutions for (2+1)-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov equation and coupled Schrödinger-KdV equation.

On the similarity solutions of the semi-linear hyperbolic equationu xt =f(u) via symmetry method

Herein the physically and mathematically significant semi-linear hyperbolic equationuxt= f(u), where f(u) is an arbitrary smooth function ofu and which encompasses the Liouville equation, Phi-four equation, Sine-Gordon equation, Klein-Gordon equation, Mikhailov equation and double Sine-Gordon equation, has been analysed via the symmetry method as developed by Steinberg (Symmetry methods in differential equations, Technical Report No. 367, University of New Mexico, 1979). The infinitesimals, similarity variables, dependent variables and reduction to quadrature or exact solutions have been tabulated for the mentioned physical forms of f(u). Some interesting outcomes of this study are the deductions of the new exact solutions that does not seem to have been reported in the literature.

The Miura transformations of Kaup's equation and of Mikhailov's equation

Two types of Miura transformations have been found. One relates Kaup's equation to Sawada-Kotera's equation and another relates the generalized Mikhailov equation to a model equation for a shallow water wave.

Chelates of o-quinone—with Cu(II) and Fe(III) acting as hydroxylators and oxidators. Organic catalysts: Catalytic activity of o-quinones

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.