New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

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.     

Exact solutions of the Duffin-Kemmer-Petiau equation with a hyperbolical potential in (1+3) dimensions

We study S-wave solutions of the Duffin-Kemmer-Petiau (DKP) equation in the presence of a hyperbolical potential in (1+3)-dimensional space-time for spin-one particles. The exact analytical Nikiforov-Uvarov (NU) method is used in the calculations to obtain the eigenfunctions and the corresponding eigenvalues. Some figures and numerical values are included to give a better insight to the solutions.     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

Modified KdV equation for solitary Rossby waves with β effect in barotropic fluids

This paper uses the weakly nonlinear method and perturbation method to deal with the quasi-geostrophic vorticity equation, and the modified Korteweg-de Vries(mKdV) equations describing the evolution of the amplitude of solitary Rossby waves as the change of Rossby parameter β(y) with latitude y is obtained.     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein－Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.     

解的移植法与含源耦合VCmKdV方程的解

根据变系数modified Korteweg-de Vries（VCmKdV）方程与常系数KdV-mKdV方程的非线性项、色散项的相似性,对解已知的KdV-mKdV方程做适当变换,并将它的解移植到解未知的VCmKdV方程,由此构造出两个不同方程解之间的移植关系.利用这种解的移植方法,求得了由两层流体模型经演化获得的含有源（或汇）耦合VCmKdV系统新的精确解和类孤波解.对Bcklund变换与解的移植法进行了比较,分析了源和汇对波幅的影响. 关键词： 解的移植法 KdV-mKdV方程 耦合VCmKdV系统 类孤波解     

A Coupled Hybrid Lattice: Its Related Continuous Equation and Symmetries

The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.     

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation.     

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.     

两类非线性方程的精确解

利用行波约化方法，并借助于一维立方非线性Klein-Gordon方程的精确解，求出了（1+1）维Zakharov方程组、变系数Korteweg-de Vries方程的一些精确解- 关键词： 行波约化方法 一维立方非线性Klein-Gordon方程 （1+1）维Zakharov方程组 变系数Korteweg-de Vries方程     

Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method

In this Letter, the Fan sub-equation method is used to construct exact solutions of a generalized Hirota-Satsuma coupled KdV equation. Many exact traveling wave solutions are successfully obtained, which contain more general solitary wave solutions and Jacobian elliptic function solutions with double periods. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

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

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

Breather solutions of a negative order modified Korteweg-de Vries equation and its nonlinear stability

This paper is concerned with a negative order modified Korteweg-de Vries (nmKdV) equation which is in the negative flow of the mKdV hierarchy. We construct the breather solutions by Hirota's bilinear method and derive the infinite conservation laws through the Lax pair of the nmKdV equation. By constructing a new Lyapunov function with the conservation laws, we obtain the nonlinear stability of the breather solutions.