From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two‐component extended Kadomtsew‐Petviashvili (KP) hierarchy. As a by‐product, the N‐soliton solutions in the form of determinants for these equations are provided.     

Davey-Stewartson系列的无色散化探讨

从Davey-Stewartson(DS)系列的定义和性质出发,按照标准的无色散方法,研究了DS系列的无色散(半经典)极限.利用整个系列的特征函数的WKB表示,在无色散极限下,得到了无色散DS系列的哈密顿-雅可比型的非线性Lax表示,给出了无色散DS系列的流方程.最后,通过两个例子验证了结论的正确性.     

Davey-Stewartson系列的无色散化探讨

从Davey-Stewartson(DS)系列的定义和性质出发,按照标准的无色散方法,研究了DS系列的无色散(半经典)极限.利用整个系列的特征函数的WKB表示,在无色散极限下,得到了无色散DS系列的哈密顿-雅可比型的非线性Lax表示,给出了无色散DS系列的流方程.最后,通过两个例子验证了结论的正确性.     

无色散p-约化KP系列的弦方程

无色散p-约化KP系列是无色散KP系列的约化形式.基于两个形式劳伦级数,由特殊的附加对称流,无色散p-约化KP系列的弦方程被给出.     

Rotating Loop Soliton of the Coupled Dispersionless Equations

We investigate soliton solutions of the coupled dispersionless equations that describe a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space with bilinear equations. We obtain a new type of loop soliton solutions that rotate around the Z axis. We also investigate the two-soliton interaction.     

Some Higher Order Complex Vector Functional Equations

In the present paper some complex vector functional equations of higher order without parameters and with complex parameters are solved.     

Adiabatic Limit for Some Nonlinear Equations of Gauge Field Theory

We consider the adiabatic limit for nonlinear dynamic equations of gauge field theory. Our main example of such equations is given by the Abelian (2+1)-dimensional Higgs model. We show next that the Taubes correspondence, which assigns pseudoholomorphic curves to solutions of Seiberg--Witten equations on symplectic 4-manifolds, may be interpreted as a complex analogue of the adiabatic limit construction in the (2+1)-dimensional case.     

Meromorphic Solutions of Some Complex Non-Linear Difference Equations

Analysis Mathematica - This paper is devoted to investigating the growth and zeros of meromorphic solutions of the generalized Fermat functional equation $${f^2}\left({z + 1} \right) + {A_1}\left(z...     

On the Reduction of Some Dispersionless Integrable Systems

We study the invariance, reduction, exact solutions and conservation laws of the dispersionless Kadomtsev-Petviashivili and the heavenly equation. The existence of nontrivial conservation laws lead to repeated reductions paving the way for determining exact solutions. A variety of such solutions exist dependent on the subalgebra of Lie point symmetries that is chosen in the reduction procedure.     

一类复差分方程亚纯解的振荡性质

该文利用Nevanlinna理论讨论了一类复差分方程亚纯解的振荡性质,包括零点和极点分布、增长级、以及方程可能的退化形式,其中方程系数与亚纯解的特征函数之间满足一定关系,推广了郑秀敏和陈宗煊的相关结果.     

一些数学物理问题中的Hamilton方程

讨论了新的一系列在数学物理方程中微分方程的Hamilton正则表示,其中包括变系数2阶对称方程的Hamilton系统,关于常系数的4阶对称方程新的非齐次Hamilton表示,MKdV方程以及KP方程的正则表示.     

复分析课中若干逻辑命题思考

根据我们长期从事复分析教学的经验 ,如果一个教师讲课的逻辑性强并刻意培养学生的逻辑论证的审美趣味使其认识到数学的逻辑论证对其它学科的普遍意义 ,那么这样的课程对学生是有征服力的 .即使数学系中一部分并不打算将数学作为终生职业的学生 ,或为其课程中精妙绝伦的数学悬念所陶醉 ,或醒悟到数学的思维方法对他们今后人生道路是有益的 ,因而对数学课程的学习化被动为主动 .数学知识的传授固然重要 ,而学生素质和能力的培养更重要 .在讲授本课程的过程中 ,对与课程相关的若干逻辑问题作了一些思考 ,把一些想法贯彻到教学中去 ,收到一些成…     

复变函数的可视化问题

首先简单介绍了复变函数基础理论的内容可视化和其基于数学软件的直观演示资源建设情况,然后通过幅角函数可视化问题介绍利用数学软件开发应用于数学教学实践的演示素材的方法,解决了Mathematica中的标准幅角函数不能直接表示复数幅角变化量的问题.     

非线性椭圆型复方程组的一类边值问题

本文讨论了一阶非线性椭圆型复方程组于平面多连通区域上的一类复合边值问题解的先验估计及可解性.     

On Some Inverse Problems for First Order Operator-Differential Equations

Siberian Mathematical Journal - We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in...     

Some Properties of Meromorphic Solutions to Systems of Complex Differential-Difference Equations

Applying Nevanlinna theory of the value distribution of meromorphic functions, the author studies some properties of Nevanlinna counting function and proximity function of meromorphic solutions to a type of systems of complex differential-difference equations. Specifically speaking, the estimates about counting function and proximity function of meromorphic solutions to systems of complex differential-difference equations can be given.