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

Abstract

Abstract A singular nonlinear differential equation z σ d w d z = a w + z w f ( 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 = 0 z=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 f f 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 = 0 z=0 in C {\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z → 0 w ( 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.