Generalized Riccati Equation Mapping Research Articles

Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

Exact Solutions To Biswas - Arshed Equation By Using The Extended Generalized Riccati Equation Mapping Method

Exact Solutions To Biswas - Arshed Equation By Using The Extended Generalized Riccati Equation Mapping Method

Short comment on “[formula omitted] -expansion method proposed by Manafian and Lakestani”

This comment deals with the tan (ϕ/2) -expansion method proposed recently in the two articles (Manafian and Lakestani,Optik,127,2016,2040-2054 and Manafian,Optik,127,2016,4222-4245) for solving NLPDEs. We will show by simple calculations that the tan (ϕ/2) -expansion method is exactly similar to the well-known generalized Riccati equation mapping method proposed in the article (Zhu, Chaos,Solitons and Fractals,37,2008,1335-1342). It is unlikely to propose a method which is exactly equivalent to a well-known method used previously many times for solving NLPDEs.

Structures of exact and solitary optical solutions for the higher-order nonlinear Schrödinger equation and its applications in mono-mode optical fibers

In this research, we give a new structure of solutions for the higher-order nonlinear Schrödinger equation and its applications to mono-mode optical fibers. To achieve this, we implement two recent methods, namely, the generalized Kudryashov method and generalized Riccati equation mapping method. The new optical soliton solutions for the higher-order nonlinear Schrödinger equation which refers to an optical field are obtained. The obtained solutions demonstrate the adequacy, strength, and effectiveness of both methods used and also show their ability to deal with different forms and patterns of the nonlinear partial differential equation.

The Generalized Riccati Equation Mapping for Solving (cmZKB) and (pZK) Equations

The generalized Riccati equation mapping is extended which is powerful and straight for Word mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for Combined (1+3) Zakharov-Kuznetsov-burgers equation (cmZKB) and potential (1+3) Zakharov-Kuznetsov Equation (Pzk)by applying this method. In this method, \(Q^{'}=l+nQ+mQ^2\) is used, as the auxiliary equation, called the generalized Riccati equation, where l, m and n are arbitrary constants. Further, the solutions are expressed in terms of the hyperbolic function, the trigonometric function and elliptic function.

New abundant wave solutions of the conformable space–time fractional (4+1)-dimensional Fokas equation in water waves

A nonlinear fractional model arising in water waves theory, namely the new conformable space–time fractional (4+1)-dimensional Fokas equation, is explored via some recently developed approaches, namely, the functional variable, the generalized Kudryashov, the Jacobi elliptic function expansion and the generalized Riccati equation mapping methods. As a result, several (abundant) distinct types of new exact solutions are constructed with the aid of symbolic computation via Mathematica 9 in terms of solitary wave, periodic wave, Jacobi elliptic function and other solutions. Furthermore, some graphical representations (numerical simulations) have been provided to illuminate the physical behavior of the resultant solutions that may be very important in real world problems and applications with fractional order.

Some new solutions of the Caudrey\u2013Dodd\u2013Gibbon (CDG) equation using the conformable derivative

New exact solutions of the space–time conformable Caudrey–Dodd–Gibbon (CDG) equation have been derived by implementing the conformable derivative. The generalized Riccati equation mapping method is applied to figure out twenty-seven forms of exact solutions, which are soliton, rational, and periodic ones. Also, for some suitable values of parameters, the exact solutions are found, namely dark, bell type, periodic, soliton, singular soliton, and several others, by using the conformable derivative. These types of solutions have not been proclaimed so far. 2D and 3D graphical patterns of some solutions are also given for clarification of physical features. The conformable derivative is one of the excellent choices to solve the nonlinear conformable problems arising in theory of solitons and many other areas. The results are new and very interesting for the large community of researchers working in the field of mathematics and mathematical physics.

Some new solutions of the conformable extended Zakharov-Kuznetsov equation using Atangana-Baleanu conformable derivative

The generalized Riccati equation mapping method, coupled with Atangana?s conformable derivative is implemented to solve non-linear extended Zakharov-Kuznetsov equation which results in producing hyperbolic, trigonometric and the rational solutions. The obtained results are new and are of great importance in engineering and applied sciences.

Solitons and other solutions to the nonlinear Bogoyavlenskii equations using the generalized Riccati equation mapping method

Recently, the authors of the article (Zahran and khater in Appl Math Model 40:1769–1775, 2016) have applied the modified extended tanh-function method to the nonlinear Bogoyavlenskii equations and found very few special solutions. In our article, we generalize this work by showing that the modified extended tanh-function method is just a special case of the generalized Riccati equation mapping method. Many families of exact solutions of the nonlinear Bogoyavlenskii equations have been found using the generalized Riccati equation mapping method. Bright–dark–singular soliton solutions and other solutions are obtained. Comparing our results with the well-known results are given.

Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method

In this paper, through the generalized Riccati equation mapping method, we investigate soliton solutions in the upper and lower forbidden band gab of the Salerno equation describing nonlinear discrete electrical lattice. As a result, we obtain various hyperbolic and trigonometric functions solutions and for some appropriated parameters we obtain exact solutions including kink, antikink, breathers, and dark and bright solitons. The obtained solutions are useful for the signal transmission through the electrical lattice.

Nonlinear propagation of weakly relativistic ion-acoustic waves in electron–positron–ion plasma

This work presents theoretical and numerical discussion on the dynamics of ion-acoustic solitary wave for weakly relativistic regime in unmagnetized plasma comprising non-extensive electrons, Boltzmann positrons and relativistic ions. In order to analyse the nonlinear propagation phenomena, the Korteweg–de Vries (KdV) equation is derived using the well-known reductive perturbation method. The integration of the derived equation is carried out using the ansatz method and the generalized Riccati equation mapping method. The influence of plasma parameters on the amplitude and width of the soliton and the electrostatic nonlinear propagation of weakly relativistic ion-acoustic solitary waves are described. The obtained results of the nonlinear low-frequency waves in such plasmas may be helpful to understand various phenomena in astrophysical compact object and space physics.

New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma

The theoretical and numerical studies have been investigated on the nonlinear propagation of electrostatic ion-acoustic waves (IAWs) in an un-magnetized Thomas–Fermi plasma system consisting of electron, positrons, and positive ions for both of ultra-relativistic and non-relativistic degenerate electrons. Korteweg-de Vries (K-dV) equation is derived from the model equations by using the well-known reductive perturbation method. This equation is solved by employing the generalized Riccati equation mapping method. The hyperbolic functions type solutions to the K-dV equation are only considered for describing the effect of plasma parameters on the propagation of electrostatic IAWs for both of ultra-relativistic and non-relativistic degenerate electrons. The obtained results may be helpful in proper understanding the features of small but finite amplitude localized IAWs in degenerate plasmas and provide the mathematical foundation in plasma physics.

Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method

In this paper, we investigate exact soliton solutions to a nonlinear transmission line. Using a concise and simple method known as the generalized Riccati equation mapping method, we solve a continuous nonlinear model associated with the previous nonlinear transmission line. As a result, we obtain miscellaneous travelling wave solutions including trigonometric, hyperbolic, and rational functions.

TRAVELLING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS

Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a flat bottom. The main objective in this paper is to apply the generalized Riccati equation mapping method for obtaining more exact traveling wave solutions of Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation. More precisely, the obtained solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functional form. Solutions obtained are potentially significant for the explanation of better insight of physical aspects of the considered nonlinear physical models.

Comment on " Application of Improved - Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations, Adv. Appl. Math. Mech. 4(2012) 122-130"

The authors of the above article proposed the improved ( / ) G G  expansion method and found some traveling wave solutions for each of two nonlinear evolution equations in mathematical physics, namely the Regularized Long Wave (RLW) equation and the Symmetric Regularized Long Wave (SRLW) equation. In the present article, we have noted that if we use a suitable transformation, the improved (G’/G)expansion method can be reduced into the well -known generalized Riccati equation mapping method which provides us with much more traveling wave solutions, namely twenty seven solutions for each of these two nonlinear evaluation equations. Comparison between the results of these two methods is presented. General Terms 02.30.Jr, 05.45.Yv, 02.30.Ik

Traveling wave solutions of density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method

Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.

Many Families of Exact Solutions for Nonlinear System of Partial Differential Equations Describing the Dynamics of DNA

The objective of this paper is to apply the improved generalized Riccati equation mapping method to find many families of exact traveling wave solutions for the general nonlinear dynamic system in a new double-chain model of DNA. This model consists of two long elastic homogeneous strands connected with each other by an elastic membrane. Hyperbolic and trigonometric function solutions of this model are obtained. Comparison between our results and the well-known results are given.

Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

Many exact solutions for nonlinear dynamics of DNA model using the generalized Riccati equation mapping method

The objective of this paper is to apply the generalized Riccati equation mapping method for constructing the exact traveling wave solutions of a nonlinear partial differential equation describing the dynamics of DNA modeling. This model consists of two long elastic homogeneous strands connected with each other by an elastic membrane. Hyperbolic and trigonometric function solutions of this model are obtained. Comparison between our new results and the well-known results are given. Key words: Nonlinear dynamics of DNA model, generalized Riccati equation mapping method, exact traveling wave solutions, generalized Riccati equation.

Travelling wave solutions of the generalized Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method

Exact hyperbolic, trigonometric and rational travelling wave solutions to the (2+1)-dimensional Generalized Zakharov-Kuznetsov (GZK) equation via the extended generalized Riccati equation mapping method are presented in this paper. The twenty one travelling wave solutions obtained were verified by putting them back into the GZK equation with the aid of Mathematica. This shows that the extended generalized Riccati equation mapping method is a powerful tool for finding exact solution to nonlinear partial differential equations in physics, mathematics and other applications.