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postgraduate thesis: Exact meromorphic solutions of complex algebraic differential equations
Title  Exact meromorphic solutions of complex algebraic differential equations 

Authors  
Advisors  Advisor(s):Ng, TW 
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Wong, K. [黃國堅]. (2012). Exact meromorphic solutions of complex algebraic differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833021 
Abstract  For any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science.
In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9].
Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve.
As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic SwiftHohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13].
RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function. 
Degree  Master of Philosophy 
Subject  Differentialalgebraic equations. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/173917 
HKU Library Item ID  b4833021 
DC Field  Value  Language 

dc.contributor.advisor  Ng, TW   
dc.contributor.author  Wong, Kwokkin.   
dc.contributor.author  黃國堅.   
dc.date.issued  2012   
dc.identifier.citation  Wong, K. [黃國堅]. (2012). Exact meromorphic solutions of complex algebraic differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833021   
dc.identifier.uri  http://hdl.handle.net/10722/173917   
dc.description.abstract  For any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science. In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9]. Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve. As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic SwiftHohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13]. RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function.   
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.source.uri  http://hub.hku.hk/bib/B48330218   
dc.subject.lcsh  Differentialalgebraic equations.   
dc.title  Exact meromorphic solutions of complex algebraic differential equations   
dc.type  PG_Thesis   
dc.identifier.hkul  b4833021   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4833021   
dc.date.hkucongregation  2012   