New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov Equations Using General Projective Riccati Equation Method

Applying the generalized method, which is a direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraic system, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, we can not only successfully recover the previously known travelling wave solutions found by existing various tanh methods and other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons, and periodic solutions.     

Dielectric constant of graphene-on-polarized substrate: A tight-binding model study

By using the bifurcation theory of planar dynamical systems and the qualitative theory of differential equations, we studied the dynamical behaviours and exact travelling wave solutions of the modified generalized Vakhnenko equation (mGVE). As a result, we obtained all possible bifurcation parametric sets and many explicit formulas of smooth and non-smooth travelling waves such as cusped solitons, loop solitons, periodic cusp waves, pseudopeakon solitons, smooth periodic waves and smooth solitons. Moreover, we provided some numerical simulations of these solutions.     

Explicit Exact Solutions for Some Nonlinear Partial Differential Equations with Nonlinear Terms of Any Order

In this paper, by introducing some appropriate transformation and with the help of symbolic computation, we study exact travelling wave solutions for the high-order modified Boussinesq equation, a single nonlinear reaction-diffusion equation and a generalized nonlinear Schrödinger equation with nonlinear terms of any order by use of the extended-tanh method. Thus, some new exact travelling-wave solutions, which contain kink-shaped solitons, bell-shaped solitons, periodic solutions, combined formal solitons, rational solutions and singular solitons for these equations, are obtained.     

Comparison between the generalized tanh–coth and the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ^′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.     

联立薛定谔方程的不传播光孤子和传播光孤子

映射法是一种非常经典、有效而且非常成熟的一种求解非线性演化方程的方法,其最大的特点是可以有无穷多个不同形式的设解,使得最终求得的解丰富多彩。传统的方法是在行波约化的前提下,即在常微分方程下进行映射。将这种方法进行扩展,推广成变系数的非行波约化下的映射,取得了成功,并利用改进的里卡蒂(Riccati)方程映射法,得到了联立薛定谔方程(负KdV方程)新的精确解。根据所得到的解模拟出了联立薛定谔方程的不传播光孤子(时间光孤子和亮-暗脉冲光孤子)和传播光孤子,以及光孤子的中和现象。     

Applications of the first integral method to nonlinear evolution equations

<正>In this paper,we establish travelling wave solutions for some nonlinear evolution equations.The first integral method is used to construct the travelling wave solutions of the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations.The obtained results include periodic and solitary wave solutions.The first integral method presents a wider applicability to handling nonlinear wave equations.     

Exact solutions of some physical models using the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion method

The (G′/G)-expansion method and its simplified version are used to obtain generalized travelling wave solutions of five nonlinear evolution equations (NLEEs) of physical importance, viz. the (2+1)-dimensional Maccari system, the Pochhammer–Chree equation, the Newell–Whitehead equation, the Fitzhugh–Nagumo equation and the Burger–Fisher equation. A variety of special solutions like periodic, kink–antikink solitons, bell-type solitons etc. can easily be derived from the general results. Three-dimensional profile plots of some of the solutions are also drawn.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.     

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation.In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-brightrogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

非线性离散微分方程的双曲函数法求解

本文推广了双曲函数方法用于求解非线性离散系统。求解离散的(2+1)维Toda系统和离散的mKdV系统，成功地得到了离散钟型孤立子、离散冲击波型孤立子及一些新的精确行波解。     

New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics

With the aid of Mathematica, we obtain several types of explicit and exact travelling wave solutions to a system of variant Boussinesq equations by using an improved sine-cosine method and the Wu elimination method. These solutions contain Wang's results and other types of solitary wave solutions and new solutions. The method can also be applied to solve more systems of nonlinear partial differential equations, such as the coupled KdV equations.     

Solutions of the perturbed KdV equation for convecting fluids by factorizations

In this paper, we obtain some new explicit travelling wave solutions of the perturbed KdV equation through recent factorization techniques that can be performed when the coefficients of the equation fulfill a certain condition. The solutions are obtained by using a two-step factorization procedure through which the perturbed KdV equation is reduced to a nonlinear second order differential equation, and to some Bernoulli and Abel type differential equations whose solutions are expressed in terms of the exponential andWeierstrass functions.     

Bifurcations and new exact travelling wave solutions for the bidirectional wave equations

By using the method of dynamical system, the bidirectional wave equations are considered. Based on this method, all kinds of phase portraits of the reduced travelling wave system in the parametric space are given. All possible bounded travelling wave solutions such as dark soliton solutions, bright soliton solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, numerical simulations are conducted for dark soliton solutions, bright soliton solutions and periodic travelling wave solutions to the bidirectional wave equations. The results presented in this paper improve the related previous studies.     

Exact travelling wave solutions to some fifth order dispersive nonlinear wave equations

Exact travelling wave solutions to some nonlinear equations of fifth order derivatives are derived by using some accurate ansatz methods.     

Bifurcation methods of dynamical systems for handling nonlinear wave equations

By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.      

New Exact Travelling Wave Solutions to Kundu Equation

Based on a first-order nonlinear ordinary differential equation with Six-degree nonlinear term, we first present a new auxiliary equation expansion method and its algorithm. Being concise and straightforward, the method is applied to the Kundu equation. As a result, some new exact travelling wave solutions are obtained, which include bright and dark solitary wave solutions, triangular periodic wave solutions, and singular solutions. This algorithm can also be applied to other nonlinear evolution equations in mathematical physics.