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

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC

ACS Applied Electronic Materials Pub Date : 2024-01-01 DOI:10.1515/anona-2023-0120

J. Diblík, M. Ruzicková

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引用次数: 0

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

{\mathbb{C}}

\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 ∈ C ⧹ { 0 }

a\in {\mathbb{C}}\setminus \left\{0\right\}

, and that the function

is analytic in a neighbourhood of the origin in

C × C

{\mathbb{C}}\times {\mathbb{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

查看原文本刊更多论文

奇点附近一类非线性复微分方程的消失解和炸裂解

奇异非线性微分方程 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), 其中 σ > 1 \sigma \gt 1 , 在点 z = 0 z=0 的邻域中考虑，如果 σ \sigma 是自然数，则该点位于复平面 C {\mathbb{C}} 中；如果 σ \sigma 是有理数，则该点位于有理函数的黎曼曲面中；如果 σ \sigma 是无理数，则该点位于对数函数的黎曼曲面中。假定 w = w ( z ) w=w\left(z) , a ∈ C ⧹ { 0 } a\in {\mathbb{C}}\setminus \left\{0\right\} , 并且函数 f f 是有理数。并且函数 f f 在 C × C 的原点邻域中是解析的 {\mathbb{C}}\times {\mathbb{C}}. .考虑到 σ \sigma 是整数、有理数或无理数，对于上述每一种情况，都证明了解析解 w = w ( z) 的存在性。

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来源期刊

ACS Applied Electronic Materials Multiple-

CiteScore

7.20

自引率

4.30%

发文量

567