Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method

In this paper, an extended Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the exact periodic solutions of some polynomials or nonlinear evolution equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

二维色散长波方程组的精确解

利用齐次平衡法给出了二维色散长波方程组的定态解、孤立波解与非孤立波解等几种显式精确解。这个方法也可用来寻找其它非线性发展方程的不同类型的精确解。     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

BIFURCATIONS AND NEW EXACT TRAVELLING WAVE SOLUTIONS OF THE COUPLED NONLINEAR SCHRDINGER-KdV EQUATIONS

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrdinger-KdV equations are studied. Based on this method, all phase portraits of the system in the parametric space are given. All possible bounded travelling wave solutions such as solitary wave solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, the numerical simulations are conducted for solitary wave solutions and periodic travelling wave solutions to the coupled nonlinear Schrdinger-KdV equations. The results show that the presented findings improve the related previous conclusions.     

mKdV方程和mKP方程组的新的精确孤立波解

用三角函数假设法和一种新辅助方程的解构造mK dV方程和mKP方程组的精确孤立波解.这种方法也可用于寻找其它非线性发展方程的新的孤立波解.     

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

Solitary waves of the EW and RLW equations

Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long–wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank–Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the L₂-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.     

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems.     

Nonlinear Summation of Power Series and Exact Solutions of Evolution Equations

The technique of quadratic and cubic summation of power series in the perturbation method was first used for finding exact solutions to nonlinear evolution equations. The series were construction with the use of exponential partial solutions to linearized equations. The solution of both classic and modified nonintegrable Korteweg-de Vries equations, the modified Burgers equation, and the Fisher one allows one to demonstrate specific features of the mentioned method. We obtain exact solitary wave solutions to the mentioned equations in the form of a wave impulse and a wave front and show that summation parameters depend on the pole orders of the desired solutions.     

On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.     

Solitary waves of the EW and RLW equations

Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long–wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank–Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the L₂-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

基于吴方法的孤波自动求解软件包及其应用

基于非线性代数方程组的吴特征列方法,在计算机代数系统Maple上实现了非线性微分方程孤波解的自动求解,编制了一个小型实用的软件包。作为应用,考虑了一个一般的五阶模型方程,利用该软件包获得了此方程新的孤波解以及孤子解存在的条件。     

Exact travelling wave solutions of the discrete sine–Gordon equation obtained via the exp-function method

In this paper, we generalize the exp-function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations, to nonlinear differential–difference equations (NDDEs). As an illustration, two series of exact travelling wave solutions of the discrete sine–Gordon equation are obtained by means of the exp-function method. As some special examples, these new exact travelling wave solutions can degenerate into the kink-type solitary wave solutions reported in the open literature.     

New exact solutions for mCH and mDP equations by auxiliary equation method

With the aid of symbolic computation, auxiliary equation method is introduced to investigate modified forms of Camassa-Holm and Degasperis-Procesi equations. A series of new exact traveling wave solutions, including smooth solitary wave solution, peakons, singular solution, periodic wave solution, Jacobi elliptic solution, are obtained in general form. These new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.