Meromorphic Solutions Of Some Types Of Nonlinear Complex Differential Equations

Differential equations in the complex domain is an area of mathematics admitting several ways of approach.The local theory is perhaps the most investigated of these ap-proaches.Its basic results,say the local existence and uniqueness theorem of solutions,singularity theory etc.,can be found in a large number of text-books of differential e-quations.The global theory,can also be studied in many different ways.For instance,one may consider it from the algebraic point of view,from the differential equations point of view,or from the direction of function theory.Using the Nevanlinna theory to get insight into the properties and structures of solutions of complex differential equations is from the direction of function theory.The theory of value distribution of meromorphic functions was established by R.Nevanlinna in 1925.Its first applications to the complex differential equations were made,by F.Nevanlinna[34]in 1929,who considered the differential equation f"+ A(z)f = 0 in the case of a polynomial A(z)in connection of a study of meromorphic functions with maximal deficiency sum,by R.Nevanlinna[37],who considered the same equation in connection of covering surfaces with finitely many branch points and by K.Yosida[47],who proved the celebrated Malmquist theorem via the Nevanlinna theory.The first one who made systematic studies in the applications of Nevanlinna theory into complex differential equations was H.Wittich beginning from 1942.And the article by A.Gol'dberg[11]is per-haps the most important paper treating general algebraic differential equations from earlier contributions.To the end of sixties,the global theory of complex differential equations,in connection with Nevanlinna theory,became more popular.During 1970s and 1980s,several active groups of mathematicians in different countries have played a remarkable role.In 1993,I.Laine wrote a very well book "Nevanlinna Theory and Complex Differential Equations"[21],which gave a concise treatment of Nevanlinna theory applications into the global theory of complex differential equations,covering the classical results and current research trends.Since then,a large number of con-tributions are made to this area of research.Both linear differential equations and nonlinear differential equations are considered thoroughly and fruitful results are ob-tained.In particular,it is always an interesting and quite difficult problem to prove the existence of the entire or meromorphic solutions of a given differential equation and find out the solutions if the solutions exist.In this paper,some different kinds of nonlinear differential equations are considered to find out existence conditions for meromorphic solutions and their structures.Our results extend and improve some known results obtained most recently.This paper is divided into five chapters.In Chapter 1,we introduce some basic results of the Nevanlinna theory.In Chapter 2,we firstly introduce the definition and some notions about algebraic complex differential equations;Then we introduce the Wiman-Valiron theory,which is a useful tool to consider the existence of solutions of complex differential equations;At last,two famous lemmas,the Clunie lemma and the Tumura-Clunie lemma,which are obtained by applying the Nevanlinna theory to some given differential equations,are introduced.In Chapter 3,we consider transcendental meromorphic solutions of the following two types of nonlinear differential equations:fn+Pd(f)=P1(z)e?1(z)+p2(z)e?2(z),and fnf(k)+Pd(f)=P1(z)e?1(z)+p2(z)e?2(z),where n,k,d are positive integers.In particular,we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms.In Chapter 4,we consider meromorphic solutions of the following type of Briot-Bouquet equations:a1 f'2+a2f f'+a3f2+a4f'+a5f+a6=0,where a1,a2,…,a6 are constants.In particular,we give out the conditions for the existence of meromorphic solutions and their structures.In Chapter 5,we study the following type of the Schwarzian equation:(?)where Pm(z),Qn(z)are two irreducible polynomials of degree m,resp.n satisfying m?n-2.We give a new method to find when the above equation has meromorphic solutions and the structure of solutions.