Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.     

New Generalized Transformation Method and Its Application in Higher-Dimensional Soliton Equation

A new generalized transformation method is presented to find more exact solutions of nonlinear partial differential equation. As an application of the method, we choose the (3+1)-dimensional breaking soliton equation to illustrate the method. As a result many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic function solutions, and rational solutions, are obtained. The new method can be extended to other nonlinear partial differential equations in mathematical physics.     

New Solutions to Generalized mKdV Equation

Trial function method is applied to solve generalized mKdV (GmKdV for short) equations. It is shown that GmKdV equations with a real number parameter can be solved directly by this method without a transformation, and more new kinds of solitary wave solutions are obtained.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f(ξ), is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

高阶CIP数值方法及其在相关物理问题中的应用

利用函数的高阶空间导数值构建其高次插值,得到高阶CIP(Constrained Interpolation Profile)数值算法,并在此基础上模拟研究等离子体物理中著名的伏拉索夫-泊松(Vlasov-Poisson)方程相关物理问题.高阶CIP数值方法具有更高数值精度,从而可以在同等精度的情况下减少计算格点数,加速数值计算速度.     

A Polynomial Expansion Method and New General Solitary Wave Solutions to KS Equation

Using a polynomial expansion method, the general exact solitary wave solution and singular one areconstructed for the non-linear KS equation. This approach is obviously applicable to a large variety of nonlinear evolution equation.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

Abstract In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f （ξ）, is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to (3 1)-dimensional Kadomstev-Petviashvili(KP) equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to （3＋1 ）-dimensional Kadomstev-Petviashvili（KP） equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f(ξ), is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

A Higher-Dimensional Hirota Condition and Its Judging Method

When a one-dimensional nonlinear evolution equation could be transformed into a bilinear differential form as F（Dt, Dx）f . f = O, Hirota proposed a condition for the above evolution equation to have arbitrary N-soliton solutions, we call it the 1-dimensional Hirota condition. As far as higher-dimensional nonlinear evolution equations go, a similar condition is established in this paper, also we call it a higher-dimensional Hirota condition, a corresponding judging theory is given. As its applications, a few two-dimensional KdV-type equations possessing arbitrary N-soliton solutions are obtained.     

New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions (including kinky periodic solitary-wave solutions, periodic soliton solutions, and crosskink-wave solutions) for the new (2+1)-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

New B(a)cklund Transformation and New Exact Solutions to (2+1)-Dimensional KdV Equation

A new Backlund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneousbalance method. And making use of the Backlund transformation and choosing a special seed solution, we get specialtypes of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions isconstructed, from which abundant localized coherent structures of the equation in question can be induced.     

Some New Exact Traveling Wave Solutions of Double-sine-Gordon Equation

The double-sine-Gordon equation is studied by means of the so-called mapping method. Some new exact solutions are determined.     

Elliptic Equation and Its Direct Applications to Nonlinear Wave Equations

Elliptic equation is taken as an ansatz and applied to solve nonlinear wave equations directly. More kinds of solutions are directly obtained, such as rational solutions, solitary wave solutions, periodic wave solutions and so on. It is shown that this method is more powerful in giving more kinds of solutions, so it can be taken as a generalized method.     

New Exact Solutions to the Combined KdV and mKdV Equation

The modified mapping method is developed to obtain new exact solutions to the combined KdV and mKdV equation. The method is applicable to a large variety of nonlinear evolution equations, as long as odd- and even-order derivative terms do not coexist in the equation under consideration.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f（ξ）, is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

New Baecklund Transformation and New Exact Solutions to （2＋1）-Dimensional KdV Equation

A new Baecklund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneous balance method. And making use of the Baecklund transformation and choosing a special seed solution, we get special types of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions is constructed, from which abundant localized coherent structures of the equation in question can be induced.     

New Jacobian Elliptic Function Solutions to Modified KdV Equation: I

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdV equation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's all results are obtained. When the modulus m→1 or 0, we can find the corresponding six solitary wave solutions and six trigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian elliptic function solutions and can be applied to other nonlinear differential equations.     

New Jacobian Elliptic Function Solutions to Modified KdV Equation: Ⅰ

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu‘s allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.