变系数广义KdV方程新的类孤波解和精确解

用普通KdV方程作变换，构造变系数广义KdV方程的解，获得了变系数广义KdV方程新的Jacobi椭圆函数精确解、类孤波解、三角函数解和Weierstrass椭圆函数解. 关键词： KdV方程 变系数广义KdV方程 类孤波解 精确解     

缓变地形下Rossby波振幅演变满足的带有强迫项的KdV方程

运用非线性方法与摄动法，讨论了当地形随时间缓变时Rossby波振幅的演变问题.从均值流体准地转涡度方程推导，得到Rossby波振幅演变满足带有强迫项的KdV方程的结论，而地形随时间的缓慢变化诱导了该方程的强迫项. 关键词： 非线性Rossby波 带有强迫项的KdV方程 摄动法 缓变地形     

广义随机KdV方程新的精确类孤子解

利用厄米(Hermite)变换求出了广义随机KdV方程新的类孤子解.这种方法的基本思想是通过厄米变换把Wick类型的广义随机KdV变成广义变系数KdV方程，利用特殊的截断展开方法求出 方程的解，然后通过厄米的逆变换求出方程的随机解. 关键词： 随机KdV方程 随机孤子解 白色噪音 截断展开方法 厄米变换     

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利用数值方法研究了双温离子、磁场、非均匀性和波的斜向传播对三维非线性尘埃声孤波振幅和宽度的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准变系数Korteweg-de Vries(KdV)方程。然后把非标准变系数KdV方程变为标准变系数KdV方程,并且得到了此标准变系数KdV方程的近似解析解。研究结果表明,此系统中存在着两种形式的孤波,即压缩型孤波和稀疏型孤波,外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与波的斜向传播和非均匀性有着非常紧密的联系。     

用试探方程法求变系数非线性发展方程的精确解

将试探方程法应用到变系数非线性发展方程的精确解的求解.以两类变系数KdV方程为例，在相当一般的参数条件下求得了丰富的精确解，其中包括新解. 关键词： 试探方程法 变系数KdV方程 类椭圆正弦(余弦)波解 类孤子解     

组合KdV方程的显式精确解

借助计算机代数系统Mathematica，利用双曲函数法找到了组合KdV方程(Combined KdV Equation)的精确孤立波解，包括钟型孤立波解和扭结型孤立波解.在此基础上又对双曲函数法的思想进行了推广，从而获得了其更多的显式精确解，包括间断型激波解和指数函数型解.这种方法也适用于求解其他非线性发展方程(组). 关键词： 组合KdV方程 双曲函数法 孤立波解 精确解     

非热离子对非均匀碰撞热尘埃等离子体中三维孤波的影响

从理论上研究了非热离子、外部磁场、碰撞对非均匀热尘埃等离子体中三维非线性尘埃声孤波的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准的变系数Korteweg-de Vries(KdV)方程。然后把非标准KdV方程变为标准的变系数KdV方程,并且得到了标准的变系数KdV方程的近似解析解。由此解析解可以看出,非热离子的数目、碰撞、非均匀性、波的斜向传播、尘埃颗粒和非热离子的温度对三维非线性尘埃声孤波的振幅和宽度有很大的影响。外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与非热离子、波的斜向传播、尘埃颗粒的温度和非均匀性有关。     

耦合KdV方程的约化及求解

利用Clarkson和Kruskal(CK)直接方法,对耦合KdV方程进行相似约化,同时从李群出发对该约化方程作了群论解释.进一步地,借助Ablowitz-Ramani-Segur(ARS)算法对耦合方程展开Painlevé测试,找到了3个Painlevé可积模型.最后通过形变映射法,求得耦合KdV方程的准确解析解. 关键词： 耦合KdV方程 CK直接法 Painlevé分析法 准确解析解     

改进的tanh函数方法与广义变系数KdV和MKdV方程新的精确解

利用改进的tanh函数方法将广义变系数KdV方程和MKdV方程化为一阶变系数非线性常微分方 程组-通过求解这个变系数非线性常微分方程组，获得了广义变系数KdV方程和MKdV方程新的 精确类孤子解、有理形式函数解和三角函数解- 关键词： 改进的tanh函数方法 类孤子解 有理形式函数解 三角函数解     

用影射法求解KdV方程

应用影射法求解KdV方程,得到了KdV方程的物理上有意义的两种类型的行波解,即孤立波解和Jacobi椭圆函数解,并绘图加以说明.     

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments,such as the ionosphere plasma.     

The Bi-Hamiltonian Structure and New Solutions of KdV6 Equation

We show that the KdV6 equation and the nonholonomic perturbation of bi-Hamiltonian system of KdV hierarchy recently studied in Karasu-Kalkanli et al. (J Math Phys 49:073516, 2008) and Kupershmidt (Phys Lett A 372:2634–2639, 2008) are equivalent to the Rosochatius deformation of KdV equation and KdV hierarchy with self-consistent sources (RD-KdVESCS, RD-KdVHSCS), respectively, recently presented in Yao and Zeng (J Phys A Math Theor 41:295205, 2008). The t-type bi-Hamiltonian formalisms of KdV6 equation and RD-KdVHSCS are constructed by taking x as evolution parameter. Some new solutions of KdV6 equation, such as soliton, positon and negaton solution, are presented.     

A discrete KdV equation hierarchy: continuous limit,diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.     

(3+1)维 KdV 方程新的精确解和孤子解

用普通Korteweg-de Vries(KdV)方程作变换,构造(3 1)维KdV方程的解,获得了新的孤子解、Jaoobi椭圆函数解、三角函数解和Weierstrass椭圆函数解.     

An extended functional transformation method and its application in some evolution equations

In this paper, an extended functional transformation is given to solve some nonlinear evolution equations. This function, in fact,is a solution of the famous KdV equation, so this transformation gives a transformation between KdV equation and other soliton equations. Then many new exact solutions can be given by virtue of the solutions of KdV equation.     

NEW SYMMETRIES AND LIE ALGEBRAS OF THE GENERAL KdV EQUATION

The strong symmetry of the general KdV equation is factorized to a simple form and then the inverse strong symmetry is obtained explicitly. Acting a strong symmetry of the general KdV equation on the trivial symmetry and the known re symmetry, we obtain four new sets of symmetries of the general KdV equation. All these sets of symmetries constitute an infinite dimensional Lie algebra.     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

The Bilinear Integrability,N-soliton and Riemann-theta function solutions of B-type KdV Equation

In this paper, the bilinear integrability for B-type KdV equation have been explored. According to the relation to tau function, we find the bilinear transformation and construct the bilinear form with an auxiliary variable of the B-type KdV equation. Based on the truncation form, the Bäcklund transformation has been constructed. Furthermore, the N-soliton solutions and Riemann-theta function 1-periodic solutions of the B-type KdV equation are obtained.