Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
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Date
2024-02-05
Authors
Diblík, Josef
Růžičková, Miroslava
ORCID
0000-0001-5009-316XAdvisor
Referee
Mark
Journal Title
Journal ISSN
Volume Title
Publisher
De Gruyter
Altmetrics
Abstract
A singular nonlinear differential equation z(sigma) dw/dz = aw + zwf(z , w), where sigma > 1, is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C if sigma is a natural number, in a Riemann surface of a rational function if sigma is a rational number, or in the Riemann surface of logarithmic function if sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a is an element of C { 0 } a, and that the function f f is analytic in a neighbourhood of the origin in C x C . Considering 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(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in 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 is proved and an asymptotic behaviour of w (z) s established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.
Description
Citation
Advances in Nonlinear Analysis. 2024, vol. 13, issue 1, p. 1-44.
https://www.degruyter.com/document/doi/10.1515/anona-2023-0120/pdf
https://www.degruyter.com/document/doi/10.1515/anona-2023-0120/pdf
Document type
Peer-reviewed
Document version
Published version
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Language of document
en