For some large classes of differential equations of the first order we give bounds for Ahlfors-Shimizu characteristics of meromorphic solutions in the complex plane of these equations. The considered equations largely generalize algebraic ones for which the obtained results imply the known Goldberg theorem. Characteristics of meromorphic solutions in a given domain weren't studied at al. We consider solutions in a given domain of some (large) equations and give bounds for Ahlfors-Shimizu characteristic for these solutions.

2010年以来,N.A.Kudryashov等分别对多类数理方程和复的自治常微分方程的精确解进行研究，得到了一部分亚纯解的结果。然而，我们发现对于很多数理方程组以及有应用性物理意义的偏微分方程组，运用复化法研究的还是不多。因此，研究对于许多有物理背景的高阶数理方程以及偏微分方程组如经适当变量替换复化后都可探讨其亚纯解的存在性与表现形式。及在得出解的形式后，我们将进一步分类研究并给出大量数理方程组显式精确解，运用相关数学软件进行演算来确定新的解并绘制函数解的三维动画图形，以便更好研究解的性质。本项目所研究的内容一部分是跨学科、跨方向的问题，一部分是对原有领域相关问题的继续深入。特别是若能在有应用的工程、物理背景意义下的数理方程组经适当变量替换后，探讨非线性微分方程亚纯解的存在性与表示形式以及代数微分方程增长性的估计的问题中有所突破。 Since 2010,n.a.Kudryashov et al.have studied the exact solutions of multiple mathematical equations and complex autonomous ordinary differential equations respectively,and obtained the results of partial meromorphic solutions.However,we find that for many mathematical equations and partial differential equations with practical physical meaning,the application of the complex method is still rare.Therefore,many high order mathematical equations with physical background and partial differential equations,such as the existence and representation of their meromorphic solutions,can be studied after proper variable substitution and complex.And after we get the solution form,we will further classify and study and give a large number of explicit and accurate solutions of mathematical equations,The related mathematical software performs the calculation to determine the new solution and draw the 3d animation graph of the function solution,so as to study the solution better Properties.Part of the research content of this project is interdisciplinary and cross-directional problems,and Part of it is to continue to deepen the problems in the original field.Especially if can application in engineering,physical significance under the background of mathematical equations after appropriate variable substitution,discusses the existence of meromorphic solutions of nonlinear differential equation and representation,and the growth of algebraic differential equation estimation problem of a breakthrough.

The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painleve equations 3-6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.

In this paper, the exact solutions for (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation have been investigated. By Lie group method and traveling wave transformation, we obtain two symmetry reduced equations of (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation. Then three classes of non-traveling wave exact solutions of (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation are constructed. At last, we achieve some computer simulations to illustrate our main results. © Springer International Publishing AG 2018.

In this paper, the complex method is employed to derive meromorphic solutions to a class of algebraic differential equations related to Painleve equation III, and then we illustrate our main results by some computer simulations. As an example, we obtain the meromorphic solutions of a nonlinear evolution equation by the application of our results.

In this paper, the dynamics of the functions f(mu)(z) = z exp (z + mu) with the real parameter is studied. We say that a real parameter mu belongs to the set B-n for a positive integer n if f(mu) has an attracting cycle of n-order. We prove that the Fatou set F(f(mu)) is a completely invariant attracting basin for every parameter mu < 0. Further, regarding the set Bn for n > 1, we prove the following results: (1) There exists mu(*) not equal +infinity such that B-2 = (2, mu(*)). (2) For every positive integer n > 2, the set B-n is non-empty. (3) For every prime number p > 3, the set B-p has at least two components.

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV-mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV-mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w(r,2)(z) and simply periodic solutions w(s,2)(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright (c) 2016 John Wiley & Sons, Ltd.

In this paper, we study a complete intersection surface singularity of Brieskorn type and provide a condition for the coincidence of the fundamental cycle and the minimal cycle on the minimal resolution space.

Estimating the growth of meromorphic solutions has been an important topic of research in complex differential equations. In this paper, we devoted to considering uniqueness problems by estimating the growth of meromorphic functions. Further, some examples are given to show that the conclusions are meaningful.