The Soliton Solution of （3＋1）—Dimensional Kadomtsev—Petviashvili Equations

A simple algebraic transformation relation of a special type of solution between the (3 1)-dimensional Kadomtsev-petviashvili(KP) equation and the cubic nonlinear Klein-Gordon equation (NKG) is established.Using known solutions of the NKG equation,we can obtain many soliton solutions and periodic solution of the (3 1)-dimensional KP equation.     

Solutions to Generalized mKdV Equation

A transformation is introduced for generalized mKdV (GmKdV for short) equation and Jacobi elliptic function expansion method is applied to solve it. It is shown that GmKdV equation with a real number parameter can be solved directly by using Jacobi elliptic function expansion method when this transformation is introduced, and periodic solution and solitary wave solution are obtained. Then the generalized solution to GmKdV equation deduces to some special solutions to some well~known nonlinear equations, such as KdV equation, mKdV equation, when the real parameter is set specific values.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

Exotic Localized Coherent Structures of New（2＋1）－Dimensional Soliton Equation

The variable separation approach is used to obtain localized coherent structures of the new(2 1)-dimensional nonlinear partial differential equation.Applying the Baecklund transformation and introducing the arbitrary functions of the seed solutions,the abundance of the localized structures of this model are derived.Some special types of solutions solitoff,dromions,dromion lattice,breathers and instantons are discussed by selecting the arbitrary functions appropriately .The breathers may breath in their amplititudes,shapes,distances among the peaks and even the number of the peaks.     

Exact solution and exotic coherent solition structures of the （2＋1）—dimensional generalized nonlinear Schroedinger equation

In this paper,a variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized nonlinear Schroedinger equation:Iφt-(α-β）φxx (α β）φyy-2λφ[（α β）（∫-∞^x|φ|y^2ydx u1(y,t))-(α-β）(∫-∞^y|φ|x^2dy u2(x,t))]=0,By applying a special Baecklund transformation and introducing arbitrary functions of the seed solutions,the abundance of the localized structures of this model are derived.By selecting the arbitrary function appropriately,some special types of localized excitations such as dromions,dromion lattice,breathers and istantons are constructed.     

Exact Solutions of （2＋1）-Dimensional Euler Equation Found by Weak Darboux Transformation

The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.     

Solution of Gauss-Codazzi Equation with Applications in the Tzitzeica Equation

The surface in R3 associated with the Tzitzeica equation ＆ considered. By curvature coordinate transformation and surface imbedding, the Gauss-Codazzi equation is presented. Resorting to the solutions of the Gauss-Codazzi equation, the solution of the Tzitzeica equation ＆ obtained under a restrictive condition.     

Explicit N—Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrodinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrodinger equations is constructed with the help of a gauge transformation of spectral problems.As a reduction,the Darboux transformation for well-known Gerdjikov-Ivanov equation is further obtained,from which a general form of N-soliton solutions for Gerdjikov-Ivanov equation is given.     

Searching for the （3＋1）—Dimensional Painleve Integrable Model and its Solitary Wave Solution

A （3 1)-dimensional integrable model constructed by conformal invariants is proven to be integrable.The solitary wave solution of the model is obtained by a simple algebraic transformation relation between the (3 1)-dimensional Harry-Dym equation and the cubic nonlinear Klein-Gordon equation.     

Painleve Property,Beicklund Transformations and Rouge Wave Solutions of （3 ＋ 1）-Dimensional Burgers Equation

Burgers equation is the simplest one in soliton theory, which has been widely applied in almost all the physical branches. In this paper, we discuss the Painleve property of the （3＋1）-dimensional Burgers equation, and then Becklund transformation is derived according to the truncated expansion of the obtained Painleve analysis. Using the Backlund transformation, we find the rouge wave solutions to the equation via the multilinear variable separation approach. And we aiso give an exact solution obtained by general variable separation approach, which is proved to possess abundant structures.     

The D'Alembert type waves and the soliton molecules in a (2+1)-dimensional Kadomtsev-Petviashvili with its hierarchy equation

For a one (2+1)-dimensional combined Kadomtsev-Petviashvili with its hierarchy equation, the missing D'Alembert type solution is derived first through the traveling wave transformation which contains several special kink molecule structures. Further, after introducing the Bäcklund transformation and an auxiliary variable, the N-soliton solution which contains some soliton molecules for this equation, is presented through its Hirota bilinear form. The concrete molecules including line solitons, breathers and a lump as well as several interactions of their hybrid are shown with the aid of special conditions and parameters. All these dynamical features are demonstrated through the different figures.     

sine-Gordon型方程的无穷序列新解

通过下列步骤,获得了sine-Gordon型方程的新解.第一步、通过函数变换,把sine-Gordon方程与sinhGordon方程的求解问题转化为两种非线性常微分方程的求解问题.第二步、获得了两种非线性常微分方程与第一种椭圆方程的拟B?cklund变换.第三步、利用第一种椭圆方程的B?cklund变换与新解,构造了sine-Gordon型方程的无穷序列新解.     

Bäcklund Transformat ion and Mult isoliton-Like Solutions for (2+1)-Dimensional Dispersive Long Wave Equations

A Bäcklund transformation of the (2+1)-dimensional dispersive long wave equations is derived by using the developed homogeneous balance method. by means of the Bäcklund transformation, the new multisoliton-like solution and other two types of exact solutions to these equations are constructed.     

Solitons on a Periodic Wave Background of the Modified KdV-Sine-Gordon Equation

The Bäcklund transformation(BT) of the mKdV-sG equation is constructed by introducing a new transformation. Infinitely many nonlocal symmetries are obtained in terms of its BT. The soliton-periodic wave interaction solutions are explicitly derived by the classical Lie-group reduction method. Particularly, some special concrete soliton and periodic wave interaction solutions and their behaviours are discussed both in analytical and graphical ways.     

A New (2+1)-Dimensional KdV Equation and Its Localized Structures

A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique. Exact solutions of the new equation are studied by means of the singular manifold method. Bäcklund transformation in terms ofthe singular manifold is obtained. And localized structures are also investigated.     

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.     

New Exact Solution of (N+1)-Dimensional Burgers System

In this letter, using a Bäcklund transformation and the new variable separation approach, we find a new general solution of the (N+1)-dimensional Burgers system. The form of the universal formula obtained from many (2+1)-dimensional system is extended.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

A Bäcklund Transformation of the Restricted mKdV Flow with a Rosochatius Deformation

A Bäcklund transformation of the restricted mKdV flow with a Rosochatius deformation is constructed. Its Lax representation and thus N invariants in involution are presented. Such Bäcklund transformation is a Rosochatius deformation of that of the restricted mKdV flow.