几类非线性复微分方程的亚纯解研究
发布时间：2018-09-01 18:03
【摘要】：研究复微分方程有多种方法。局部理论是研究得最多的一种方法。在许多有关复微分方程的书中都可以找到局部解的存在和唯一性定理、奇点理论等局部理论的基本结果。而全局理论也有不同的研究切入点。比如可以从代数的观点来研究,可以从微分方程的观点来研究,也可以从函数论的观点来研究。利用Nevanlinna理论研究复微分方程亚纯解的性质和结构属于函数论的研究范畴。亚纯函数的值分布理论由R.Nevanlinna于1925年创立。首先尝试将其应用于复微分方程的研究工作包括:F.Nevanlinna在1929年将其与亚纯函数最大亏量和理论结合,对微分方程f"+A(z)f=0所做的研究[34],其中A(z)为多项式;R.Nevanlinna将其与支点有限的覆盖曲面理论结合,对同样的方程所做的研究[37];以及K.Yosida利用Nevanlinna理论对著名的Malmquist定理给予的证明[47]。1942年,H.Wittich开始了早期对Nevanlinna理论应用于复微分方程的系统研究,而A.Gol'dberg对一般代数微分方程的研究结果[11]可能是早期研究工作中最重要的。此后,直到上世纪60年代末,利用Nevanlinna理论对复微分方程的全局解进行研究才变得流行起来。在随后的20年里,一些活跃的研究团队对该领域的研究发展起到了非常重要的作用。在前人工作的基础上,I.Laine在1993年前后编写了一本详细介绍Nevanlinna理论如何应用于复微分方程全局解研究的专著《Nevanlinna理论和复微分方程》[21],内容涵盖经典研究结果以及最新研究趋势,给有志于该领域研究的后来者提供了一个非常好的学习平台。这之后,大批该领域的研究文章和专著涌现,线性和非线性微分方程都得到了充分的研究,获得了丰硕的成果。作为其中的一个研究方向,如何证明给定的微分方程存在亚纯解,能否给出解可能具有的形式,引起了许多学者的兴趣。这篇论文,在前人研究的基础之上,针对几种类型的非线性复微分方程,给出了亚纯解的存在条件,并求得了解的结构,改进和推广了前人的结果。全文一共分为五章。第一章主要介绍Nevanlinna理论的经典结果。第二章首先介绍了代数复微分方程的基本概念和定义符号;其次介绍了Wiman-Valiron理论,该理论是研究复微分方程亚纯解是否存在的一个有力工具;最后介绍了两个将Nevanlinna理论应用于复微分方程所获得的经典结论:Clunie引理和Tumura-Clunie引理。第三章考虑了如下两类非线性微分方程和超越亚纯解的存在情况,给出了亚纯解的存在条件和可能具有的形式。在这里,n,k,d均为正整数。第四章考虑了 一种特殊类型的Briot-Bouquet方程超越亚纯解的存在情况,给出了亚纯解的存在条件和可能具有的形式。在这里,a1,a2,…,a6均为常数。第五章研究了如下类型的施瓦茨方程其中Pm(z),Qn(z)分别为m次和n次的不可约多项式,满足m≤n-2。我们找到了一种新的确定该类方程是否存在亚纯解的方法并给出了解的形式。
[Abstract]:There are many ways to study complex differential equations. Local theory is one of the most studied methods. In many books on complex differential equations, the existence and uniqueness theorems of local solutions and the basic results of local theories such as singularity theory can be found. The global theory also has different research entry points. For example, it can be studied from the point of view of algebra, from the point of view of differential equation, or from the point of view of function theory. The properties and structure of meromorphic solutions of complex differential equations are studied by Nevanlinna theory. The value distribution theory of meromorphic functions was founded by R.Nevanlinna in 1925. The first attempt to apply it to complex differential equations involves the combination of the maximum deficiency and the theory of meromorphic functions by: F. Nevanlinna in 1929. A study of the differential equation f "A (z) f ~ 0 [34], where A (z) is a polynomial and R. Nevanlinna combines it with the theory of covering surfaces with finite fulcrum. The study of the same equation [37] and the proof given by K.Yosida to the famous Malmquist theorem by using the Nevanlinna theory [47]. In 1942 H. Wittich began the early systematic study of the application of Nevanlinna theory to complex differential equations. The results of A.Gol'dberg 's study on general algebraic differential equations [11] are probably the most important in the early research work. Thereafter, it became popular to study the global solutions of complex differential equations by using Nevanlinna theory until the late 1960s. Over the next 20 years, a number of active research teams have played an important role in the development of this field. On the basis of previous work, I. Laine wrote a monograph, "Nevanlinna Theory and complex differential equation" [21], about the application of Nevanlinna theory to the global solution of complex differential equations in 1993, which covers the classical research results and the latest research trends. It provides a very good learning platform for the latecomers who wish to study in this field. After this, a large number of research papers and monographs in this field have emerged, and both linear and nonlinear differential equations have been fully studied and fruitful results have been obtained. As one of the research directions, how to prove the existence of meromorphic solutions for a given differential equation and whether to give the possible form of the solution have aroused the interest of many scholars. In this paper, on the basis of previous studies, the existence conditions of meromorphic solutions for several types of nonlinear complex differential equations are given, the structure of solutions is obtained, and the previous results are improved and generalized. The full text is divided into five chapters. The first chapter mainly introduces the classical results of Nevanlinna theory. The second chapter introduces the basic concepts and definitions of algebraic complex differential equations, and then introduces Wiman-Valiron theory, which is a powerful tool to study the existence of meromorphic solutions of complex differential equations. In the end, two classical conclusions: 1. Clunie Lemma and Tumura-Clunie Lemma obtained by applying Nevanlinna theory to complex differential equations are introduced. In chapter 3, we consider the existence of two kinds of nonlinear differential equations and transcendental meromorphic solutions, and give the existence conditions and possible forms of meromorphic solutions. In this case, the number of n / k / d is all positive integers. In chapter 4, we consider the existence of transcendental meromorphic solutions for a special type of Briot-Bouquet equation, and give the conditions and possible forms of meromorphic solutions. Here's a 1, a, a 2,. A6 is constant. In chapter 5, we study the following types of Schwartz equations, where Pm (z) Q n (z) is irreducible polynomials of degree m and degree n, respectively, and satisfies m 鈮，
本文编号：2217944
[Abstract]:There are many ways to study complex differential equations. Local theory is one of the most studied methods. In many books on complex differential equations, the existence and uniqueness theorems of local solutions and the basic results of local theories such as singularity theory can be found. The global theory also has different research entry points. For example, it can be studied from the point of view of algebra, from the point of view of differential equation, or from the point of view of function theory. The properties and structure of meromorphic solutions of complex differential equations are studied by Nevanlinna theory. The value distribution theory of meromorphic functions was founded by R.Nevanlinna in 1925. The first attempt to apply it to complex differential equations involves the combination of the maximum deficiency and the theory of meromorphic functions by: F. Nevanlinna in 1929. A study of the differential equation f "A (z) f ~ 0 [34], where A (z) is a polynomial and R. Nevanlinna combines it with the theory of covering surfaces with finite fulcrum. The study of the same equation [37] and the proof given by K.Yosida to the famous Malmquist theorem by using the Nevanlinna theory [47]. In 1942 H. Wittich began the early systematic study of the application of Nevanlinna theory to complex differential equations. The results of A.Gol'dberg 's study on general algebraic differential equations [11] are probably the most important in the early research work. Thereafter, it became popular to study the global solutions of complex differential equations by using Nevanlinna theory until the late 1960s. Over the next 20 years, a number of active research teams have played an important role in the development of this field. On the basis of previous work, I. Laine wrote a monograph, "Nevanlinna Theory and complex differential equation" [21], about the application of Nevanlinna theory to the global solution of complex differential equations in 1993, which covers the classical research results and the latest research trends. It provides a very good learning platform for the latecomers who wish to study in this field. After this, a large number of research papers and monographs in this field have emerged, and both linear and nonlinear differential equations have been fully studied and fruitful results have been obtained. As one of the research directions, how to prove the existence of meromorphic solutions for a given differential equation and whether to give the possible form of the solution have aroused the interest of many scholars. In this paper, on the basis of previous studies, the existence conditions of meromorphic solutions for several types of nonlinear complex differential equations are given, the structure of solutions is obtained, and the previous results are improved and generalized. The full text is divided into five chapters. The first chapter mainly introduces the classical results of Nevanlinna theory. The second chapter introduces the basic concepts and definitions of algebraic complex differential equations, and then introduces Wiman-Valiron theory, which is a powerful tool to study the existence of meromorphic solutions of complex differential equations. In the end, two classical conclusions: 1. Clunie Lemma and Tumura-Clunie Lemma obtained by applying Nevanlinna theory to complex differential equations are introduced. In chapter 3, we consider the existence of two kinds of nonlinear differential equations and transcendental meromorphic solutions, and give the existence conditions and possible forms of meromorphic solutions. In this case, the number of n / k / d is all positive integers. In chapter 4, we consider the existence of transcendental meromorphic solutions for a special type of Briot-Bouquet equation, and give the conditions and possible forms of meromorphic solutions. Here's a 1, a, a 2,. A6 is constant. In chapter 5, we study the following types of Schwartz equations, where Pm (z) Q n (z) is irreducible polynomials of degree m and degree n, respectively, and satisfies m 鈮，
本文编号：2217944
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