Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

三类非线性演化方程新的Jacobi椭圆函数精确解

应用修正影射法分别求解三类非线性演化方程,即非线性Klein-Gordon方程,mKdV方程和广义Boussinesq方程,得到了一些新的Jacobi椭圆函数展开解,包括Jacobi椭圆函数解、混合Jacobi椭圆函数解、孤子解和三角函数解. 关键词： Klein-Gordon方程 mKdV方程 广义Boussinesq方程 Jacobi椭圆函数     

Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis- Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

Exact Solutions to Short Pulse Equation

In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.     

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.     

有弱偏置磁场及含时激光场中的非线性Gross-Pitaevskii方程新的精确解

利用映射方法和一个适当的变换，得到大量的有弱偏置磁场及含时激光场中的非线性Gross-Pitaevskii方程的新解，这些解包括椭圆函数解，椭圆函数叠加解，三角函数解，亮孤子解，暗孤子解和类孤子解。     

(2+1)维Broer-Kaup方程的局域分形结构

进一步拓广齐次平衡法的应用，研究了(2+1)维Broer-Kaup方程的局域分形结构.根据齐次平衡原则，得到方程的B?cklund变换，将方程变换为一个线性化的方程，然后由具有两个 任意函数的种子解构造出一个精确解.利用Jacobian椭圆函数得到了特殊的分形结构. 关键词： 齐次平衡法 (2+1)维Broer-Kaup方程 分形结构     

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.     

Peakons and compactons on the background of periodic wave

In this paper, the extended tanh-function method (ETM) based on the mapping equation is further improved by generalizing the Riccati equation. The new variable separation solutions of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system are derived. From the periodic wave solution and by selecting appropriate functions, the evolutional behaviours of peakons and compactons on the background of Jacobian elliptic wave are investigated.     

Two classes of fractal structures for the （2＋1）-dimensional dispersive long wave equation

Using the mapping approach via a Riccati equation, a series of variable separation excitations with three arbitrary functions for the (2+1)-dimensional dispersive long wave (DLW) equation are derived. In addition to the usual localized coherent soliton excitations like dromions, rings, peakons and compactions, etc, some new types of excitations that possess fractal behaviour are obtained by introducing appropriate lower-dimensional localized patterns and Jacobian elliptic functions.     

用推广的Jacobian椭圆函数方法解离散的mKdV格子

本文用推广的Jacobian椭圆函数方法求解了离散的mKdV格子，得到了Jacobian椭圆函数双周期解，当模取1时，可得到钟型孤波解和冲击型孤波解。     

一般变换下双Jacobi椭圆函数展开法及应用

将行波变换下修正的双Jacobi椭圆函数展开法推广到范围广泛的一般函数变换下进行.利用这一方法求得了一类非线性方程更多新的周期解，这些解包括了在行波变换下所求得的周期解. 关键词： Jacobi椭圆函数展开法 非线性发展方程 函数变换 周期解     

Moduli in N = 1 heterotic/F-theory duality

The moduli in a 4D N = 1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into “base” parameters which are even (under the natural involution of the elliptic curves), and “fiber” or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5-brane/F-theoretic 3-brane impurities.     

两组新的广义的Ito方程组的多种行波解

利用推广的形变映射法，得到两组广义的Ito方程组的丰富的行波解，包括孤子解，三角函数解，椭圆函数解，幂函数解. 关键词： 广义的Ito方程 非线性Klein-Gordon方程 行波解     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: Ⅱ

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by usingour extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and anotherthree families of new doubly periodic solutions (Jacobian elliptic function solutions) are fbund again by using a newtransformation, which and our extended Jacobian elliptic function expansion method form a new method still called theextended Jacobian elliptic function expansion method. The new method can be more powertul to be applied to othernonlinear differential equations.     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: II

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by using our extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and another three families of new doubly periodic solutions (Jacobian elliptic function solutions) are found again by using a new transformation, which and our extended Jacobian elliptic function expansion method form a new method still called the extended Jacobian elliptic function expansion method. The new method can be more powerful to be applied to other nonlinear differential equations.     

Quasi-Periodic Structures Based on Symmetrical Lucas Function of(2+1)-Dimensional Modified Dispersive Water-Wave System

By introducing the Lucas--Riccati method and a linear variable separationmethod, new variable separation solutions with arbitrary functions arederived for a (2+1)-dimensional modified dispersive water-wave system. Themain idea of this method is to express the solutions of this system aspolynomials in the solution of the Riccati equation that the symmetricalLucas functions satisfy. From the variable separation solution and byselecting appropriate functions, some novel Jacobian elliptic wave structurewith variable modulus and their interactions with dromions and peakons are investigated.     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: Ⅲ

More recently, sixteen families of Jacobian elliptic function solutions of mKdV equation have been foundby using our extended Jacobian elliptic function expansion method. In this paper, we continue to improve our methodby using another eight pairs of the closed Jacobian elliptic functions. The mKdV equation is chosen to illustrate theimproved method such that another eight families of new Jacobian elliptic function solutions are obtained again. Thenew method can be more powerful to be applied to other nonlinear differential equations.     

Exact Solutions Expressible in Rational Formal Hyperbolic and Elliptic Functions for Nonlinear Differential-Difference Equation

A new approach is presented by means of a new general ansätz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, discretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.     

Jacobian elliptic Chebyshev rational maps

We introduce the Jacobian elliptic Chebyshev rational maps with the the equidistributivity property (or briefly EDP) and the semi-group property like the Chebyshev polynomial maps. Simple methods are also discussed for generating sequences of i.i.d. p-ary random variables based on the Jacobian elliptic Chebyshev rational maps. Furthermore, we give conjectures on correlational properties of a real-valued trajectory generated by the Jacobian elliptic Chebyshev rational maps.