Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

Abstract

In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth are at most two. Moreover, for the second order factorizable algebraic ODEs, all the meromorphic solutions of them (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

Meromorphic solutions of the seventh-order KdV equation by using an extended complex method and Painlev? analysis

“…Ng and Wu [20] investigated the second-order nonlinear Loewy factorizable algebraic ordinary differential equations (ODEs) and showed that the conjecture proposed by Hayman in 1996 holds for some certain second-order ODEs. Although many methods for constructing solutions of ordinary differential equations have made enormous progress [21], higherorder nonlinear ordinary differential equations are rarely investigated, especially analytical solutions.…”

Singularity methods for meromorphic solutions of differential equations

“…Remark 1. Using this method, one can prove that for a large class of n-th order autonomous algebraic ODEs, their meromorphic solutions must be elliptic or degenerate elliptic (see [31]). The method fails if there is a positive Fuchs index and hence one cannot conclude the finiteness of the number of Laurent series.…”

Rationality of meromorphic functions between real algebraic sets in the plane

“…These are the only meromorphic functions which satisfy the addition formulae. They are also the only meromorphic solutions of certain non-linear complex differential equations (see [10], [4], [5], [6], [19] and [12]).…”