Search

This paper is devoted to developing a nonlocal and time-delayed reaction–diffusion model which makes contributions to understanding the mechanism of block effect on HCV by high-mobility group box 1 (HMGB1). Then basic reproduction number R 0 is established. The existence of HMGB1 can make R 0 < 1 , that is, can induce the extinction of virus. Further, threshold dynamics are investigated in terms of R 0 in a bounded spatial domain. From the biological perspective, threshold-type result can predict whether the disease will die out or persist. To investigate the invasion speed of the infectious disease, the existence of traveling waves is investigated in an unbounded spatial domain. Our results imply that the nonlocal time delay cannot change threshold dynamics, while it can slow down the wave speed of virus. [ABSTRACT FROM AUTHOR]

In this article, the modified direct algebraic method has been extended to celebrate the new complex travelling wave solutions of nonlinear evolution equations of fractional order. For finding solutions, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. After this, the modified direct algebraic method has been applied successfully, to construct the complex travelling wave solutions. As far as concerned about the applications, the nonlinear time–space fractional KdV, foam drainage and coupled burgers’ equations of fractional order have been considered. [ABSTRACT FROM AUTHOR]

In this paper, the travelling wave solutions of a nonlinear Schrödinger equation is considered by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple software, The possible explicit parametric representations of the bounded travelling wave solutions are got and all kinds of phase portraits in the parametric space are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa–Holm and Degasperis–Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made. [Copyright &y& Elsevier]

Abstract: By using the -expansion method proposed recently, we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper. The travelling wave solutions are expressed by three types of functions. The computation for the method appears to be easier and faster by general mathematical software. [Copyright &y& Elsevier]

Abstract: We present an alternative functional variable method or briefly called FVM (in short) to analyze a class of wave equations, consisting in construction of exact travelling solutions. Some important nonlinear model equations which arise in a wide variety of physical problems have been presented as testbed to show the application of this method. [Copyright &y& Elsevier]

Abstract: By using the bifurcation theory of planar dynamical systems to the nonlinear dispersion Drinfel’d–Sokolov () system, the existence of solitary wave solutions, kink and anti-kink wave solutions, compacton solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the (2+1)-dimensional Boiti–Leon–Pempinelli (BLP) equation will be studied. The tanh–coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures. [ABSTRACT FROM AUTHOR]

Abstract: The -expansion method is introduced to construct more general exact travelling wave solutions to the generalized KdV–mKdV equation with any-order nonlinear terms. As a result, hyperbolic function solution, trigonometric function solution and rational solution with parameters are obtained. When the parameters take up special values, the solitary waves are also derived from the travelling wave solutions. The -expansion method presents a wider applicability for handling nonlinear wave equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new modified Novikov equation is introduced. Multiple exact travelling wave solutions are obtained by using the extended-tanh function method. Furthermore, based on the homogeneous balance method, an auto-Bäcklund transformation is derived and some solitary wave solutions to this new equation are established. [ABSTRACT FROM AUTHOR]

Abstract: In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods. [Copyright &y& Elsevier]

Abstract: Here we find exact solutions of some nonlinear evolution equations within the framework of the -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. [Copyright &y& Elsevier]

Abstract: The higher-order nonlinear Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using a simple transformation, various solutions of the nonlinear ODE are obtained by the (G′/G)-expansion method proposed recently. With the aid of solutions of the nonlinear ODE more explicit travelling wave solutions involving two arbitrary parameters of the higher-order nonlinear Schrödinger equation are found out. The travelling waves are expressed by the hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values the solitary waves are also derived from the travelling waves. [Copyright &y& Elsevier]

Abstract: With the aid of symbolic computation and two auxiliary ordinary differential equations, the new generalized algebraic method is extended to the generalized Zakharov–Kuznetsov (GZK) equation with higher-order nonlinear terms for constructing a series of new and more general travelling wave solutions. Because of the higher-order nonlinear terms, the equation can not be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we transform the GZK equation to an ordinary differential equation that is easy to solve and find a rich variety of new exact travelling wave solutions for the GZK equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, combined Jacobi elliptic function solutions and rational function solutions. The method used here can be also extended to other nonlinear partial differential equations with higher-order nonlinear terms. [Copyright &y& Elsevier]

Abstract: In this paper, we established travelling wave solutions of the nonlinear equation. The first integral method was used to construct travelling wave solutions of the Cahn–Allen equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed. [Copyright &y& Elsevier]

Abstract: In this paper, we establish exact solutions for a nonlinear evolution equation. The sech method and the exp-function method are used to construct the solitary travelling wave solutions of Lax’s seventh-order KdV equation. These solutions may be important of significance for the explanation of some practical physical problem. [Copyright &y& Elsevier]

Abstract: By using the -expansion proposed recently the travelling wave solutions involving parameters of the Broer–Kaup equations and the approximate long water wave equations are found out. The travelling wave solutions are expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values the solitary wave solutions are obtained. [Copyright &y& Elsevier]

Abstract: An extended method is proposed for constructing new forms of exact travelling wave solutions to nonlinear evolution equations by making a more general transformation. For illustration, we apply the extended method to Whitham–Broer–Kaup equation and two-dimensional perturbed KdV equation and successfully cover the previously known travelling wave solutions found by Chen’s method [Chaos, Solitons and Fractals 22 (2004) 675; Int. J. Mod. Phys. C 4 (2004) 595]. Moreover, some new forms of solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this work, exact travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations are obtained. The analysis rests mainly on the standard tanh method. The work emphasizes the need for a transformation formula for the case where the parameter M is non-integer. The approach can be used in a variety of many types of nonlinearity. [Copyright &y& Elsevier]

Abstract: Generalized forms of the KdV, the nonlinear heat conduction, and Burgers–Fisher equations are investigated. The analysis rests mainly on the standard tanh method. The focus is on the case where the parameter M is noninteger. A variety of exact travelling wave solutions of distinct physical structures are formally derived. [Copyright &y& Elsevier]

New exact travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh-function method. The obtained results include rational, periodic, singular and solitary wave solutions. [Copyright &y& Elsevier]

In this paper, by means of a proper transformation and symbolic computation, we study the exact travelling wave solutions for a generalized Zakharov–Kuznetsov (GZK) equation by using the extended-tanh method and direct assumption method. As a result, rich exact travelling wave solutions, which contain new kink-shaped solitons, bell-shaped solitons, periodic solutions, combined formal solitons, rational solutions and singular solitons for GZK equation, are obtained. [Copyright &y& Elsevier]