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.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

A Generalized Algebraic Method for Constructing a Series of Explicit Exact Solutions of a (1+1)-Dimensional Dispersive Long Wave Equation

Making use of a new and more general ansatz, we present the generalized algebraic method to uniformlyconstruct a series of new and general travelling wave solution for nonlinear partial differential equations. As an applicationof the method, we choose a (1 1)-dimensional dispersive long wave equation to illustrate the method. As a result, wecan successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003)17] and find other new and more general solutions at the same time, which include polynomial solutions, exponentialsolutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrassdoubly periodic wave solutions.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

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.     

New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation

In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.     

Symbolic Computation and Construction of Soliton－Like Solutions to the（2＋l）－Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Mapte, a new generalized expansion method of Riccati equation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new general ansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = O,uv=vx, and obtain rich new families of the exact solutions of the breaking sofiton equation, including then on-traveilin~ wave and constant function sofiton-like solutions, singular soliton-like solutions, and triangular function solutions.     

New Soliton-like Solutions for (2+1)-Dimensional Breaking Soliton Equation

The (2+1)-dimensional breaking soliton equation describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. In this paper, with the aid of symbolic computation, six kinds of new special exact soltion-like solutions of (2+1)-dimensional breaking soliton equation are obtained by using some general transformations and the further generalized projective Riccati equation method.     

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m→1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.     

Symbolic Computation and Construction of Soliton-Like Solutions to the(2+1)-Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Maple, a new generalized expansion method of Riccatiequation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new generalansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = 0, uy = vx, and obtain rich new families of the exact solutions of the breaking soliton equation, including thenon-travelling wave and constant function soliton-like solutions, singular soliton-like solutions, and triangular functionsolutions.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the (2 1)-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.     

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.     

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.     

An extended Jacobi elliptic function rational expansion method and its application to ()-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz, in which periodic solutions of nonlinear partial differential equations that can be expressed as a finite Laurent series of some of 12 Jacobi elliptic functions, is more powerful than exiting Jacobi elliptic function methods and is very powerful to uniformly construct more new exact periodic solutions in terms of rational formal Jacobi elliptic function solution of nonlinear partial differential equations. As an application of the method, we choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.     

Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko

Based on the Hirota bilinear operators and their generalized bilinear derivatives, we formulate two new (2+1)-dimensional nonlinear partial differential equations, which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schiff equation and the generalized Bogoyavlensky-Konopelchenko equation as particular examples, and the other has the same bilinear form with different $D_p$-operators. A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches. A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.     

Reduction and New Explicit Solutions of (2+1)-Dimensional Breaking Soliton Equation

By applying the Lie group method, the (2+1)-dimensional breaking soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.     

Multisoliton Solutions of the (2 1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2 1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2 1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.``