Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The B\"{a}cklund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

Novel Solutions of KdV Equation

Some novel solutions of the KdV equation are obtained through the modified bilinear B\"{a}cklund transformation.     

Periodic folded waves for a （2＋1）-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

Folded localized excitations in the （2＋1）-dimensional modified dispersive water-wave system

By using a mapping approach and a linear variable separation approach, a new family of solitary wave solutions with arbitrary functions for the (2+1)-dimensional modified dispersive water-wave system (MDWW) is derived. Based on the derived solutions and using some multi-valued functions, we obtain some novel folded localized excitations of the system.     

A Bilinear B(a)cklund Transformation and Lax Pair for a (1+1)-Dimensional Differential-Difference sine-Gordon Equation

In this paper, we obtain a 1 1 dimensional integrable differential-difference model for the sine-Gordon equation by Hirota's discretization method. A bilinear B(a)cklund transformation and the associated Lax pair are also proposed for this model.     

Negatons, Positons, and Complexiton Solutions of Higher Order for a Non-isospectral KdV Equation

In this paper, negatons, positons, and complexiton solutions of higher order for a non-isospectral KdV equation, the KdV equation with loss and non-uniformity terms are obtained through the bilinear B(a)cklund transformation.Further, the properties of some solutions are shown by some figures made by using Maple.     

变形耦合发电机系统中的混沌控制

分别利用反馈和非反馈方法研究了变形耦合发电机系统的混沌控制问题，并基于Lyapunov直接法和Routh-Hurwitz判据讨论了受控变形耦合发电机系统的混沌轨道达到不稳定平衡点或极限环时的条件，同时给出了理论上的证明.数值模拟进一步验证了这两种方法的有效性. 关键词： 变形耦合发电机系统 反馈法 非反馈法 混沌控制     

220 GHz折叠波导返波管束流光学系统设计和实验研究

本文介绍了220 GHz折叠波导行波管电子束流光学系统的设计过程。总体来讲,该系统可分为收敛性皮尔斯电子枪、磁聚束段和单极降压收集极。对于一个工作在太赫兹频段的电真空器件而言,极其细长的束流孔道让电子注以较高的流通率穿过慢波结构变得十分困难。空间电荷效应,加工装配精度和热初速等原因都是限制流通率的重要原因。研发一个具有足够流通率的实用束流光学系统对于220 GHz折叠波导返波管的研制是迫切的且十分棘手的任务。通过理论方法和数值工具,系统的三个部分将先后被设计,以满足束波互作用分析提出的电子注要求。基于这样的设计和开展的误差分析,流通管样管成功封管并进行了初步测试。实验数据表明这样的束流光学系统可以产生15.4 kV,22 mA的电子注,并能以80%以上的流通率通过直径0.19 mm,长30 mm的束流管道。     

(2+1)维Korteweg-de Vries方程的复合波解及局域激发

借助Maple符号计算软件,利用Pdccati方程(ζ′=a_0+a_1ζ+a_2ζ~2)展开法和变量分离法,得到了(2+1)维Korteweg-de Vries方程(KdV)包含q=C_1x+C_2y+C_3t+R(x,y,t)的复合波解,根据得到的孤立波解,构造出KdV方程新颖的复合波裂变和复合波湮灭等局域激发结构。     

海-气耦合动力系统的改进变分迭代解法

研究了一个描述厄尔尼诺和南方涛动振荡物理机理的海-气耦合动力系统.利用改进变分迭代方法 (MVIM)简捷地得到了该非线性模型近似解的展开式.通过与特殊情形下模型精确解的比较,说明获得的MVIM 近似解具有非常好的准确度.     

一类非线性耦合系统的复合型双孤子新解

首先给出一种函数变换,把一类非线性耦合系统化为两个第一种椭圆方程组.然后利用第一种椭圆方程的新解与B?cklund变换,构造了一类非线性耦合系统的无穷序列复合型双孤子新解.     

A Bäcklund transformation between two integrable discrete hungry systems

The discrete hungry Toda (dhToda) equation and the discrete hungry Lotka-Volterra (dhLV) system are known as integrable discrete hungry systems. In this Letter, through finding the LR transformations associated with the dhToda equation and the dhLV system, we present a Bäcklund transformation between these integrable systems.     

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs,Bäcklund transformations,residual symmetries and exact solutions

ABSTRACT

A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.     

Recursion Formulae for Obtaining Surfaces with Constant Mean Curvature in R2,1

Though the B(a)cklund transformation on time-like surfaces with constant mean curvature surfaces in R2,1 has been obtained,it is not easy to obtain corresponding surfaces because the procedure of solving the related integrable system cannot be avoided when the B(a)cklund transformation is used.For sake of this,in this article,some special work is done to reform the B(a)cklund transformation to a recursion formula,by which we can construct time-like surfaces with constant mean curvature form known ones just by quadrature procedure.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

A kind of extended Korteweg——de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio $\varepsilon $, represented by the ratio of amplitude to depth, and the dispersion ratio $\mu $, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin {\it et al} in the study of the surface waves when considering the order up to $O(\mu ^2)$. As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin {\it et al} for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.     

An improved method for Darboux transformation for the coupled inhomogeneous nonlinear Schrödinger system from plasma physics and nonlinear optics

The coupled inhomogeneous nonlinear Schrödinger-type system which can be used to control soliton propagation and interaction in certain plasmas and optical fibers is investigated. An improved method for Darboux transformation (DT) is presented in more general forms by constructing an improved Γ$\Gamma$-Riccati-type Bäcklund transformation (Γ$\Gamma$-R BT). With the N$N$th-iterated Γ$\Gamma$-R BT or the N$N$th-iterated DT, which is a compact representation for the N$N$-soliton-like solutions and can generate a series of analytic solutions from a pair of the seed solutions through algebraic manipulations, the analytic one-/two-soliton-like solutions are provided. With the choice of parameters for the soliton solutions, the dynamical characteristics of the influences of the inhomogeneous parameters on the propagation of the soliton pulses are discussed graphically.     

不变本征算符法计算各向异性海森伯亚铁磁系统的自旋波能量

利用不变本征算符法计算了X-Y-Z模型各向异性海森伯亚铁磁系统的自旋波能量,并讨论了此系统特殊情形下的自旋波能量及不变本征算符法的优点与不足.     

Comparison between two different sliding mode controllers for a fractional-order unified chaotic system

Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.