Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point

Abstract

A singular nonlinear differential equation zσdwdz=aw+zwf(z,w),{z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ>1\sigma \gt 1, is considered in a neighbourhood of the point z=0z=0 located either in the complex plane C{\mathbb{C}} if σ\sigma is a natural number, in a Riemann surface of a rational function if σ\sigma is a rational number, or in the Riemann surface of logarithmic function if σ\sigma is an irrational number. It is assumed that w=w(z)w=w\left(z), a∈C⧹{0}a\in {\mathbb{C}}\setminus \left\{0\right\}, and that the function ff is analytic in a neighbourhood of the origin in C×C{\mathbb{C}}\times {\mathbb{C}}. Considering σ\sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w=w(z)w=w\left(z) in a domain that is part of a neighbourhood of the point z=0z=0 in C{\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property limz→0w(z)=0{\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w(z)w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

Keywords

• analytic solution
• asymptotic behaviour
• blow-up phenomenon
• complex plane
• differential equation
• singular point
• 34m35
• 34m30
• 34m10
• 34a25