(2+1)维色散长波方程新的类孤子解

通过一个简单的变换,将(2+1)维色散长波方程简化为人们熟知的带强迫项Burgers方程,借助Mathematica软件,利用齐次平衡原则和变系数投影Riccati方程法,求出了(2+1)维色散长波方程新的精确解.     

(2+1)维Kadomtsev-Petviashvili方程的留数对称及其相互作用解

运用Painlevé截断展开方法得到(2+1)维Kadomtsev-Petviashvili(KP)方程的非局域留数对称和B?cklund变换.由于非局域对称不能直接对(2+1)维KP方程进行约化求解,因此,需要将非局域对称局域化.然后,利用相容的Riccati展开(CRE)可解的概念证明(2+1)维KP方程的CRE可解性,从而求出(2+1)维KP方程的新的相互作用解.     

(2+1)维色散长波方程的非局域对称及相容Riccati展开可积性

研究了(2+1)维色散长波方程的非局域对称性和相容Riccati展开(CRE)可积性.首先,通过Painleve分析中的留数对称,将(2+1)维色散长波方程留数对称局域化,得到了与Schwartzian变量相对应的对称群;其次,基于CRE方法,证明了(2+1)维色散长波方程在CRE条件下是可积的;最后,通过求解相容性方程,构造了该方程的孤立波与椭圆周期波的相互作用解.     

(2+1)-维变系数CDGKS方程的Bcklund变换及Gramm-type Pfaffian解

孤立子在非线性的流体力学、等离子物理学、光学、生物学等领域有广泛的应用.将(2+1)维常系数CDGKS方程扩展为(2+1)维变系数CDGKS方程,利用双线性方法求出了该方程的Bcklund变换,进一步求出变系数CDGKS方程及其修正变系数CDGKS方程的Gramm-type Pfaffian解,从而解决了变系数孤立子方程的精确解.     

(2+1)维耦合MKP型方程的分解及其显式解

利用(2+1)维耦合MKP型方程与它分解后的(1+1)维DNLS方程之间的关系,用达布变换的方法求出(1+1)维DNLS方程的显式解,进而得到(2+1)维耦合MKP型方程的显式解.     

Riccati方程与(2+1)维非线性物理模型的分离变量解

对于(2+1)维非线性物理模型,欲获其解并非易事.本文试图在变量分离法中,通过设定的R iccati方程来得到方程的解.同时,以(2+1)维Bo iti-L eon-M anna-P em p inelli方程和Burgers方程为例来说明之.     

一类非线性发展方程的对称分析及级数解

基于李对称理论分析了广义Burgers方程的推广方程,获得其有限维李对称.进一步,研究向量场的伴随表示构造优化系统.最终基于对称约化,获得了方程的约化系统及包含级数解在内的群不变解.     

两个高维loop代数及应用

借助于循环数,构造了维数分别是5(s+1)和4(s+1)的两个高维loop代数．为了计算方便,本文只考虑s=1时的应用．利用第一个loop代数■_1~*得到了具有4-Hamilton结构的一个广义AKNS族,该方程族可约化为著名的AKNS族．利用第二个loop代数■_2~*,得到了具有4个分量位势函数的4-Hamilton结构方程族,该族可约化为一个非线性耦合Burgers方程和一个耦合的KdV方程．     

利用Hirota双线性法求解一类非线性发展方程

利用hirota双线性法和Hopf-Cole变换,得到(3+1)维广义KP方程、广义(3+1)维浅水波方程、(1+1)维Boussinesq方程、(2+1)维Nizhnik方程的精确解,并做出一部分解的图形,进一步研究解的结构和性质.实践证明,方法对于研究非线性发展方程具有十分重要的作用.     

(2+1)维广义破裂孤子方程的非局域对称及相互作用解

根据截断的Painlevé分析展开法及相容Riccati展开(CRE)法,研究了(2+1)维广义破裂孤子方程的非局域对称.利用非局域对称局域化的方法,得到了与Schwarzian变量相对应的对称群.同时,证明了这个方程是CRE可积的,并给出了它的孤立波与椭圆周期波之间的相互作用解.     

New origins of hidden symmetries for partial differential equations

Hidden symmetries of differential equations are point symmetries that arise unexpectedly in the increase (equivalently decrease) of order, in the case of ordinary differential equations, and variables, in the case of partial differential equations. The origins of Type II hidden symmetries (obtained via reduction) for ordinary differential equations are understood to be either contact or nonlocal symmetries of the original equation while the origin for Type I hidden symmetries (obtained via increase of order) is understood to be nonlocal symmetries of the original equation. Thus far, it has been shown that the origin of hidden symmetries for partial differential equations is point symmetries of another partial differential equation of the same order as the original equation. Here we show that hidden symmetries can arise from contact and nonlocal/potential symmetries of the original equation, similar to the situation for ordinary differential equations.     

Nonlocal symmetries,exact solutions and conservation laws of the coupled Hirota equations

Using the Lax pair, nonlocal symmetries of the coupled Hirota equations are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are successfully localized to Lie point symmetries. With the help of Lie symmetries of the closed prolongation, exact solutions and nonlocal conservation laws of the coupled Hirota equations are studied.     

High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.     

A variational formulation approach to a generalized coupled inhomogeneous Emden–Fowler system

We study a generalized coupled inhomogeneous Emden–Fowler system from the Lagrangian formulation standpoint. A special case of this system was considered in the literature, and necessary and sufficient conditions for the existence of multiple positive solutions were obtained. Here we perform preliminary Noether classification of the generalized system by the direct method. We obtain nine cases for which the system has Noether point symmetries. First integrals are then obtained for the cases which admit Noether point symmetries.     

Symmetry analysis and conservation laws of the geodesic equations for the Reissner-Nordström de Sitter black hole with a global monopole

This paper is devoted to the comprehensive analysis of the problem of symmetries and conservation laws for the geodesic equations of the Reissner-Nordström de Sitter (RNdS) black hole with a global monopole. For this purpose, the system of geodesic equations is determined and the corresponding classical Lie point symmetry operators are obtained. An optimal system of one dimensional subalgebras is constructed and a brief discussion about the algebraic structure of the Lie algebra of symmetries is presented. Also, the Noether symmetries of the geodesic Lagrangian is calculated. Finally, by applying two methods including Noether’s theorem and direct method the conservation laws associated to the system of geodesic equations are obtained.     

Exact solutions of KdV equation with time-dependent coefficients

In this paper, we study the Korteweg-de Vries (KdV) equation having time dependent coefficients from the Lie symmetry point of view. We obtain Lie point symmetries admitted by the equation for various forms for the time-dependent coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary coefficients. Subsequently, the 1-soliton solution is obtained by the aid of solitary wave ansatz method. It is observed that the soliton solution will exist provided that these time-dependent coefficients are all Riemann integrable.     

Nonlocal symmetries and exact solutions of the (2+1)-dimensional generalized variable coefficient shallow water wave equation

In this paper, using the standard truncated Painlevé analysis, the Schwartzian equation of (2+1)-dimensional generalized variable coefficient shallow water wave (SWW) equation is obtained. With the help of Lax pairs, nonlocal symmetries of the SWW equation are constructed which be localized by a complicated calculation process. Furthermore, using the Lie point symmetries of the closed system and Schwartzian equation, some exact interaction solutions are obtained, such as soliton–cnoidal wave solutions. Corresponding 2D and 3D figures are placed to illustrate dynamic behavior of the generalized variable coefficient SWW equation.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

P ∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operators

The symmetry algebraP _∞=W _∞ ⊕P ⊕I _∞ of integrable systems is defined. As an example, the classical Lie point symmetries of all higher Kadomtsev-Petviashvili equations are obtained. It is shown that of the point symmetries, the (positive) ones belong to theW _∞ symmetries, while the other (negative) ones belong toI _∞ symmetries. The corresponding action on the τ-function is obtained for the positive symmetries. The negative symmetries cannot be obtained from the free fermion algebra. A new embedding of the Virasoro algebra intogl(∞) describes the conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 231–260, November, 1997.     

Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations

A systematic investigation to derive Lie point symmetries to time fractional generalized Burgers as well as Korteweg–de Vries equations is presented. Using the obtained Lie point symmetries we have shown that each of them has been transformed into a nonlinear ordinary differential equation of fractional order with a new independent variable. The derivative corresponding to time fractional in the reduced equation is usually known as the Erdélyi–Kober fractional derivative.