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postgraduate thesis: Meromorphic solutions of complex differential equations
Title  Meromorphic solutions of complex differential equations 

Authors  
Advisors  Advisor(s):Ng, TW 
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Wu, C. [吳成發]. (2014). Meromorphic solutions of complex differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5317034 
Abstract  The objective of this thesis is to study meromorphic solutions of complex algebraic ordinary differential equations (ODEs). The thesis consists of two main themes. One of them is to find explicitly all meromorphic solutions of certain class of complex algebraic ODEs. Since constructing explicit solutions of complex ODEs in general is very difficult, the other theme (motivated by the classical conjecture proposed by Hayman in 1996) is to establish estimations on the growth of meromorphic solutions in terms of Nevanlinna characteristic function.
The tools from complex analysis that will be used have been collected in Chapter 1. Chapter 2 is devoted to introducing a method, which was first used by Eremenko and later refined by Conte and Ng, to give a classification of some complex algebraic autonomous ODEs. Under certain assumptions, based on local singularity analysis and Nevanlinna theory, this method shows that all meromorphic solutions of these ODEs if exist, must belong to ‘class W’, which consists of elliptic functions and their degenerations. Combined with knowledge from function theory, as shown by Demina and Kudryashov, it further allows us to find all of them explicitly and the details of the method will be illustrated by constructing new real meromorphic solutions of the stationary case of cubicquintic SwiftHohenberg equation. In Chapter 3, the same method is used to construct on R^n, n ≥ 2 some explicit Bryant solitons and on R^n\{0}, n ≥ 2 some Ricci solitons, and one of them turns out to be a new Ricci soliton on R^5\{0}. In addition, the completeness of corresponding metrics on the Ricci solitons that we have constructed are also discussed.
In 1996, Hayman conjectured an upper bound on the growth, in terms of Nevanlinna characteristic function, of meromorphic solutions of complex algebraic ODEs. Related work in the literature towards this socalled classical conjecture is first reviewed in Chapter 4. The classical conjecture for three types of second order complex algebraic ODEs will then be verified by either giving a classification of the meromorphic solutions or obtaining them explicitly in Chapter 4. As the classical conjecture seems to be out of reach at present, we proposed in Chapter 5 to study a particular class of complex algebraic ODEs which can be factorized into certain form. On one hand, for these factorizable ODEs, it has been proven for the generic case that all their meromorphic solutions must be elliptic functions or their degenerations. On the other hand, the second order factorizable ODEs have been carefully studied so that their meromorphic solutions have been obtained explicitly except one case. This will allow the classical conjecture for most of the second order factorizable ODEs to be verified by employing Nevanlinna theory and certain qualitative results from complex differential equations. Finally, the classical conjecture has been shown to be sharp in certain cases. 
Degree  Doctor of Philosophy 
Subject  Differential equations 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/206466 
HKU Library Item ID  b5317034 
DC Field  Value  Language 

dc.contributor.advisor  Ng, TW   
dc.contributor.author  Wu, Chengfa   
dc.contributor.author  吳成發   
dc.date.accessioned  20141031T23:15:58Z   
dc.date.available  20141031T23:15:58Z   
dc.date.issued  2014   
dc.identifier.citation  Wu, C. [吳成發]. (2014). Meromorphic solutions of complex differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5317034   
dc.identifier.uri  http://hdl.handle.net/10722/206466   
dc.description.abstract  The objective of this thesis is to study meromorphic solutions of complex algebraic ordinary differential equations (ODEs). The thesis consists of two main themes. One of them is to find explicitly all meromorphic solutions of certain class of complex algebraic ODEs. Since constructing explicit solutions of complex ODEs in general is very difficult, the other theme (motivated by the classical conjecture proposed by Hayman in 1996) is to establish estimations on the growth of meromorphic solutions in terms of Nevanlinna characteristic function. The tools from complex analysis that will be used have been collected in Chapter 1. Chapter 2 is devoted to introducing a method, which was first used by Eremenko and later refined by Conte and Ng, to give a classification of some complex algebraic autonomous ODEs. Under certain assumptions, based on local singularity analysis and Nevanlinna theory, this method shows that all meromorphic solutions of these ODEs if exist, must belong to ‘class W’, which consists of elliptic functions and their degenerations. Combined with knowledge from function theory, as shown by Demina and Kudryashov, it further allows us to find all of them explicitly and the details of the method will be illustrated by constructing new real meromorphic solutions of the stationary case of cubicquintic SwiftHohenberg equation. In Chapter 3, the same method is used to construct on R^n, n ≥ 2 some explicit Bryant solitons and on R^n\{0}, n ≥ 2 some Ricci solitons, and one of them turns out to be a new Ricci soliton on R^5\{0}. In addition, the completeness of corresponding metrics on the Ricci solitons that we have constructed are also discussed. In 1996, Hayman conjectured an upper bound on the growth, in terms of Nevanlinna characteristic function, of meromorphic solutions of complex algebraic ODEs. Related work in the literature towards this socalled classical conjecture is first reviewed in Chapter 4. The classical conjecture for three types of second order complex algebraic ODEs will then be verified by either giving a classification of the meromorphic solutions or obtaining them explicitly in Chapter 4. As the classical conjecture seems to be out of reach at present, we proposed in Chapter 5 to study a particular class of complex algebraic ODEs which can be factorized into certain form. On one hand, for these factorizable ODEs, it has been proven for the generic case that all their meromorphic solutions must be elliptic functions or their degenerations. On the other hand, the second order factorizable ODEs have been carefully studied so that their meromorphic solutions have been obtained explicitly except one case. This will allow the classical conjecture for most of the second order factorizable ODEs to be verified by employing Nevanlinna theory and certain qualitative results from complex differential equations. Finally, the classical conjecture has been shown to be sharp in certain cases.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.subject.lcsh  Differential equations   
dc.title  Meromorphic solutions of complex differential equations   
dc.type  PG_Thesis   
dc.identifier.hkul  b5317034   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5317034   