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 cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.     

Numerical Complexiton Solutions of Complex KdV Equation

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions： numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.     

Painleve Property and Complexiton Solutions of a Special Coupled KdV Equation

A special coupled KdV equation is proved to be the Painleve property by the Kruskal＇s simplification of WTC method. In order to search new exact solutions of the coupled KdV equation, Hirota＇s bilinear direct method and the conjugate complex number method of exponential functions are applied to this system. As a result, new analytical eomplexiton and soliton solutions are obtained synchronously in a physical field. Then their structures, time evolution and interaction properties are further discussed graphically.     

Analytic Solutions to Forced KdV Equation

The Korteweg-de Vries equation with a forcing term is established by recent studies as a simple mathematical model of describing the physics of a shallow layer of fluid subject to external forcing. In the present paper, we study the analytic solutions to the KdV equation with forcing term by using Hirota＇s direct method. Several exact solutions are given as examples, from which one can see that the same type soliton solutions can be excited by different forced term.     

From Rosochatius System to KdV Equation

The Rosochatius system on the sphere, an integrable mechanical system discovered in the nineteenth century, is investigated in a suitably chosen framework with the sphere as an invariant set, to avoid the complicated constraint presentations. Higher order Rosochatius flows are defined and straightened out in the Jacobi variety of the associated hyperelliptic curve. A relation is found between these flows and the KdV equation, whose finite genus solution is calculated in the context of the Rosoehatius hierarchy.     

Wronskian Form of N-Solitonic Solution for a Variable-Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.     

Similarity Reductions and Exact Solutions of a Coupled Korteweg de Vries System

For a special coupled Korteweg de Vries （KdV） system, its similarity solutions and reduction equations are obtained by the Clarkson and Kruskal＇s direct method. In addition, its new explicit soliton solutions and traveling wave solutions are found by the deformation and mapping method.     

Effects of Adiabatic Dust Charge Fluctuation and Particles Collisions on Dust-Acoustic Solitary Waves in Three-Dimensional Magnetized Dusty Plasmas

Taking into account the combined effects of the external magnetic field, adiabatic dust charge fluctuation and collisions occurring between the charged dust grains and neutral gas particles （dust-neutral collisions）, the dust-acoustic solitary waves in three-dimensional uniform dusty plasmas are investigated analytically. By using the reductive perturbation method, the Korteweg-de Vries （KdV） equation governing the dnst-aconstic solitary waves is obtained. The present analytical results show that only rarefactive solitary waves exist in this system. It is also found that the effects of the wave vector along the z-direction, dust charge variation, collisional frequency, the plasma density, and temperature ratio can significantly influence the characteristics of low-frequency wave modes. Moreover, for the collisional dusty plasmas, there is a certain critical value μc of the plasma density ratio μ, if μ 〈 μc, the width of the waves increases with μ, otherwise the width of waves decreases with μ.     

Similarity Reductions of Nearly Concentric KdV Equation

Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries （ncKdV） equation can be reduced to three types of （1＋1）-dimensional variable coefficients partial differential equations （PDEs） and three types of variable coefficients ordinary differential equation. Furthermore, three types of （1＋1）-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.     

Gap solitons in parity time complex superlattice with dual periods

A theory is presented to investigate the existence and propagation stability of gap solitons in a parity-time （PT） com- plex superlattice with dual periods. In this superlattice, the real and imaginary parts are both in the form of superlattices with dual periods. In the self-focusing nonlinearity, PT solitons can exist in the semi-infinite gap. However, only those gap solitons with low powers can propagate stably, whereas the high-power solitons present periodic oscillation and simultane- ously suffer energy decay. In the self-defocusing nonlinearity, PT solitons only exist in the first gap and all these solitons are stable.     

Solitary and Periodic Waves in Coupled KdV Equations with Different Linear Dispersion Relations

Based on the invariant expansion method, some reasonable approximate solutions of coupled Korteweg-de Vries （KdV） equations with different linear dispersion relations have been obtained. These solutions contain not only bell type soliton solutions but also periodic wave solutions that expressed by Jacob/elliptic functions. The results also show that if the arbitrary constants are selected suitably, the approximate solutions may become the exact ones.     

Algebraic-Geometric Solution to （2＋1）-Dimensional Sawada-Kotera Equation

Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.     

（G′/G）-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.     

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. Bgcklund transformation in terms of the singular manifold is obtained. And localized structures are also investigated.     

Exact Solutions and Coherence Structures of Generalized Breor-Kaup Equations

Using the mapping method based on q-deformed hyperbolic functions, the exact solutions of generalized Breor-Kaup equations are obtained. Based on the solutions, two coherent structures, periodic-branch kink and nonpropagating kink, have been obtained. Moreover, one solitonal interaction form, two line solitons interaction on the kink background, has been found.     

An eighth-order KdV-type equation in (1+1) and (2+1) dimensions: multiple soliton solutions

In this work we study an eighth-order KdV-type equations in (1+1) and (2+1) dimensions. The new equations are derived from the KdV6 hierarchy. We show that these equations give multiple soliton solutions the same as the multiple soliton solutions of the KdV6 hierarchy except for the dispersion relations.     

Cusped Solitons and Loop-Solitons of Reduced Ostrovsky Equation

The qualitative theory of differential equations is applied to the Ostrovsky equation. The cusped soliton and loop-soliton solutions of the Ostrovsky equation are obtained. Asymptotic behavior of eusped soliton solutions is given. Numerical simulations are provided for cusped solitons and so-called loop-solitons of the Ostrovsky equation.     

微扰Burgers-KdV方程的孤子解

对于一个其耗散项可看作微扰的Burgers-KdV(B-KdV)方程u_t+uu_x+βu_xxx=εu_xx,|ε|?1,考虑一级近似和行波情形,建立一套求通解的直接扰动方法,利用零级方程的单孤子解,获得一级方程的孤子型通解,它包含任意多个不同的孤子解,每个孤子解分别描述一个位于半无限空间的孤子阵列,分析表明,耗散使得“亮孤子”变矮变窄,“暗孤子”变浅变窄. 关键词：     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

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.