The Representations Of Meromorphic Solutions To Some Higher Order Algebraic Differential Equations And Applications

In this paper,we mainly discuss the following three algebraic differential e-quations:The complex higher order KdV equationw????+?ww??+?w^?2+?w³+?w+?=0,where,?,?,?=0,??0,are constants.The complex higher order non-linear differential equation w?5?+aw?+bw²-cw+d=0,where a,b?=0,c,d are constants.The Fermat type equation afⁿ+b?f??^m?1,where n.m are positive integers,a,b,c?C,ab?0.The main work is to find representations of meromorphic solutions of the for-mer algebraic differential equations.In the first chapter we introduce the objects of study,related research advances,research aims.In the second chapter we review the fundamental knowledge of algebraic differential equations,outline advances in meromorphic solutions of some higher order differential equations and introduce the complex method.In the third chapter we discuss the expressions for mero-morphic solutions of the higher order KdV equation and give the meromorphic solutions?such as rational function solutions,simply periodic function solutions,elliptic function solutions?by the calculative software and the complex method.In the forth chapter we study the expressions and applications for meromorphic solu-tions of the higher order non-linear differential equation by the complex method.In the fifth chapter we study the Fermat type equation.Then,we propose some open questions.