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.     

Painlevé Property and Allowed Transformations of the Variable-Coefficient Zakharov-Kuznetsov Equation

In this letter, we discuss the Painlevé property and allowed transformation for the variablecoefficient Zakharov-Kuznetsov equation which governs nonlinear ion-acoustic wqves in a magnetized plasma. The general solution of the singular manifold equation and stability solution of the VCZK equation are obtained. To prove some information of the integrability of the ZK equation, we prove the constraints that the variable-coefficient functions for the equation to possess the Painlevé property are not equivalent in order that the equations may be transformed into the constant coefficient equation. So we confirm that the ZK equation is not integrable.     

Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg--de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painlev\'e I transcendent function.     

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.     

Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation

The (4 + 1)-dimensional Fokas equation is derived in the process of extending the integrable Kadomtsev–Petviashvili and Davey–Stewartson equations to higher-dimensional nonlinear wave equations. This equation is under investigation in this paper. Hirota’s bilinear method is, for the first time, used to solve such a higher-dimensional equation. In order to bilinearize the Fokas equation, some appropriate transformations are adopted. As a result, single-soliton solution, double-soliton solution and three-soliton solution are obtained. A new uniform formula of n-soliton solution is derived from this. It is shown that the transformations adopted in this work play a key role in converting the Fokas equation into Hirota’s bilinear form.     

Abundant Multisoliton Structures of the Generalized Nizhnik-Novikov-Veselov Equation

Using the extended homogenous balance method, we obtain abundant exact solution structures of a (2+1)-dimensional integrable model, the generalized Nizhnik-Novikov-Veselov equation. By means of the leading order term analysis, the nonlinear transformations of generalized Nizhnik-Novikov-Veselov equation are given first, and then some special types of single solitary wave solution and the multisoliton solutions are constructed.     

光纤中变系数非线性Schr(o)dinger方程的孤子解及其应用

基于推广的立方非线性Klein—Gordon方程对一般形式的变系数非线性Schrodinger方程进行研究，讨论了无啁啾情形的孤子解，发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解．同时对基本孤子的色散控制方法进行了简单讨论．作为特例，常系数非线性Schrodinger方程和两类特殊的变系数非线性Schrodinger方程的结果和已知的形式一致．此外，还研究了一个周期增益或损耗的光纤系统，得到了有意义的结果．     

Cylindrical KP-Burgers Equation for Two-Temperature Ions in Dusty Plasma with Dissipative Effects and Transverse Perturbations

In this paper, the cylindrical KP-Burgers equation with variable coefficient for two-temperature ions in unrsagnified dusty plasma with dissipative effects and transverse perturbations in cylindrical geometry is derived by using the standard reductive perturbation technique. With the help of variable-coeiffcient generalized projected Ricatti equation expansion method, the cylindrical KP-Burgers equation is solved and shock wave solution is obtained. The effecta of some important parameters to the shock wave solution are illustrated from the wave evolution figures. The effects caused by dissipation and transverse perturbations are also discussed.     

Cylindrical KP-Burgers Equation for Two-Temperature Ions in Dusty Plasma with Dissipative Effects and Transverse Perturbations

In this paper, the cylindrical KP-Burgers equation with variable coefficient for two-temperature ions in unmagnified dusty plasma with dissipative effects and transverse perturbations in cylindrical geometry is derived by using the standard reductive perturbation technique. With the help of variable-coefficient generalized projected Ricatti equation expansion method, the cylindrical KP-Burgers equation is solved and shock wave solution is obtained. The effects of some important parameters to the shock wave solution are illustrated from the wave evolution figures. The effects caused by dissipation and transverse perturbations are also discussed.     

Abundant Multisoliton Structure of the (3+1)-Dimensional Nizhnik-Novikov-Veselov Equation

Using the extended homogeneous balance method, we obtained abundant exact solution structures of the (3+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation. By means of the leading order term analysis, the nonlinear transformations of the (3+1)-dimensional NNV equation are given first, and then some special types of single solitary wave solution and the multisoliton solutions are constructed.     

ABUNDANT EXACT SOLUTION STRUCTURES OF THE NIZHNIK－NOVIKOV－VESELOV EQUATION

Using the extended homogeneous balance method, we have obtained abundant exact solution structures of a (2+1)-dimensional integrable model, the Nizhnik--Novikov--Veselov equation. By means of leading order terms analysis, the nonlinear transformations of the Nizhnik--Novikov--Veselov equation are given first, and then some special types of single solitary wave solution and multisoliton-like solutions are constructed.     

一维Tonks-Girardeau原子气区域中 Gross-Pitaevskii方程简化模型的精确行波解

利用动力系统方法研究一维Tonks-Girardeau原子气区域中Gross-Pitaevskii (GP)方程简化模型的一些精确行波解以及这些精确行波解的动力学行为, 研究系统的参数对行波解的动力学行为的影响. 在不同的参数条件下, 获得了一维Tonks-Girardeau原子气区域中GP方程简化模型的六个行波解的精确参数表达式. 关键词： 动力系统方法 孤立波解 周期波解 扭波解     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

(3+1)维Jimbo-Miwa方程的精确解及局域激发

在符号计算软件 Maple 的帮助下,利用投射方程法和变量分离法,得到了(3+1)维 Jimbo-Miwa(JM)方程的新显式精确解. 根据得到的孤立波解,研究了 JM 方程新颖的局域激发. 关键词： 投射方程法 Jimbo-Miwa方程 精确解 局域激发     

Finite Symmetry Transformation Groups and Exact Solutions of Konopelchenko-Dubrovsky Equation

Based on the general direct method developed by Lou et al. [J. Phys. A: Math. Gen. 38 (2005) L129], the symmetry group theorem is obtained,from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenko-Dubrovsky (KD) equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution.     

幂函数叠加势的径向薛定谔方程的解析解(英文)

当薛定谔方程中出现高次非谐振子势,电偶极矩势,分子晶体势,极化等效势等高次正幂与逆幂势函数以及它们的叠加时,薛定谔方程的求解变得非常复杂,本文采用奇点邻域附近的级数解法与求解渐近解相结合并且通过系数比较法,得到势函数为V(r)=a1r6 a2r2 a3r-4 a4r-6的径向薛定谔方程的一系列定态波函数解析解以及能级结构,并作了适当讨论与结论.     

On the extended KdV equation with self-consistent sources

The positive extended KdV equation with self-consistent sources (eKdV⁺ ESCSs) is firstly presented and its related linear auxiliary equation is derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁺ ESCSs such as N-soliton solution, N-double pole solution and nonsingular N-positon solution. The properties of these solutions are analyzed. Moreover, the interaction of two solitons is discussed in detail.     

考虑空气阻力时铅球最佳投掷角的参数方程和实用方程

指出了一些文章在讨论有空气阻力作用时关于球类运动的微分方程中存在的问题,得出了空气阻力与速度平方成正比时的微分方程及近似解,计算了最佳投掷角的参数方程,还导出了空气阻力与速度成正比时铅球最佳投掷角的实用方程.     

Soliton interaction under the influence of higher-order effects

In this paper, we present exact N-soliton solution by employing simple, straightforward Darboux transformation based on the Lax pair for Hirota equation, a higher-order nonlinear Schrödinger (HNLS) equation. As examples, one- and two-soliton solutions in explicit forms are given and their properties are also analyzed. A bound solution without interaction will be theoretically predicted if one can adjust frequency shift for each soliton appropriately. Further, we obtain the approximate eigenvalues by employing two-soliton solution and discuss analytically the interaction between neighboring solitons under the influence of the higher-order effects. It is shown that the combined effects of the higher-order effects can restrain the interaction between neighboring solitons to some extent. The results are proved by directly solving HNLS equation numerically.     

Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.