{"title":"奇点附近一类非线性复微分方程的消失解和炸裂解","authors":"J. Diblík, M. Ruzicková","doi":"10.1515/anona-2023-0120","DOIUrl":null,"url":null,"abstract":"\n <jats:p>A singular nonlinear differential equation <jats:disp-formula id=\"j_anona-2023-0120_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n <m:msup>\n <m:mrow>\n <m:mi>z</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>σ</m:mi>\n </m:mrow>\n </m:msup>\n <m:mfrac>\n <m:mrow>\n <m:mi mathvariant=\"normal\">d</m:mi>\n <m:mi>w</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi mathvariant=\"normal\">d</m:mi>\n <m:mi>z</m:mi>\n </m:mrow>\n </m:mfrac>\n <m:mo>=</m:mo>\n <m:mi>a</m:mi>\n <m:mi>w</m:mi>\n <m:mo>+</m:mo>\n <m:mi>z</m:mi>\n <m:mi>w</m:mi>\n <m:mi>f</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>z</m:mi>\n <m:mo>,</m:mo>\n <m:mi>w</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>,</m:mo>\n </m:math>\n <jats:tex-math>{z}^{\\sigma }\\frac{{\\rm{d}}w}{{\\rm{d}}z}=aw+zwf\\left(z,w),</jats:tex-math>\n </jats:alternatives>\n </jats:disp-formula> where <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>σ</m:mi>\n <m:mo>></m:mo>\n <m:mn>1</m:mn>\n </m:math>\n <jats:tex-math>\\sigma \\gt 1</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, is considered in a neighbourhood of the point <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>z</m:mi>\n <m:mo>=</m:mo>\n <m:mn>0</m:mn>\n </m:math>\n <jats:tex-math>z=0</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> located either in the complex plane <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_004.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi mathvariant=\"double-struck\">C</m:mi>\n </m:math>\n <jats:tex-math>{\\mathbb{C}}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> if <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_005.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>σ</m:mi>\n </m:math>\n <jats:tex-math>\\sigma </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is a natural number, in a Riemann surface of a rational function if <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_006.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>σ</m:mi>\n </m:math>\n <jats:tex-math>\\sigma </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is a rational number, or in the Riemann surface of logarithmic function if <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_007.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>σ</m:mi>\n </m:math>\n <jats:tex-math>\\sigma </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is an irrational number. It is assumed that <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_008.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>w</m:mi>\n <m:mo>=</m:mo>\n <m:mi>w</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>z</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>w=w\\left(z)</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_009.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>a</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"double-struck\">C</m:mi>\n <m:mo>⧹</m:mo>\n <m:mrow>\n <m:mo>{</m:mo>\n <m:mrow>\n <m:mn>0</m:mn>\n </m:mrow>\n <m:mo>}</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>a\\in {\\mathbb{C}}\\setminus \\left\\{0\\right\\}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, and that the function <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_010.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>f</m:mi>\n </m:math>\n <jats:tex-math>f</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is analytic in a neighbourhood of the origin in <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_011.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi mathvariant=\"double-struck\">C</m:mi>\n <m:mo>×</m:mo>\n <m:mi mathvariant=\"double-struck\">C</m:mi>\n </m:math>\n <jats:tex-math>{\\mathbb{C}}\\times {\\mathbb{C}}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>. Considering <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_012.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>σ</m:mi>\n </m:math>\n <jats:tex-math>\\sigma </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_013.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>w</m:mi>\n <m:mo>=</m:mo>\n <m:mi>w</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>z</m:mi>\n ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point\",\"authors\":\"J. Diblík, M. Ruzicková\",\"doi\":\"10.1515/anona-2023-0120\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n <jats:p>A singular nonlinear differential equation <jats:disp-formula id=\\\"j_anona-2023-0120_eq_001\\\">\\n <jats:alternatives>\\n <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_001.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\">\\n <m:msup>\\n <m:mrow>\\n <m:mi>z</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>σ</m:mi>\\n </m:mrow>\\n </m:msup>\\n <m:mfrac>\\n <m:mrow>\\n <m:mi mathvariant=\\\"normal\\\">d</m:mi>\\n <m:mi>w</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi mathvariant=\\\"normal\\\">d</m:mi>\\n <m:mi>z</m:mi>\\n </m:mrow>\\n </m:mfrac>\\n <m:mo>=</m:mo>\\n <m:mi>a</m:mi>\\n <m:mi>w</m:mi>\\n <m:mo>+</m:mo>\\n <m:mi>z</m:mi>\\n <m:mi>w</m:mi>\\n <m:mi>f</m:mi>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mi>z</m:mi>\\n <m:mo>,</m:mo>\\n <m:mi>w</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>,</m:mo>\\n </m:math>\\n <jats:tex-math>{z}^{\\\\sigma }\\\\frac{{\\\\rm{d}}w}{{\\\\rm{d}}z}=aw+zwf\\\\left(z,w),</jats:tex-math>\\n </jats:alternatives>\\n </jats:disp-formula> where <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_002.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>σ</m:mi>\\n <m:mo>></m:mo>\\n <m:mn>1</m:mn>\\n </m:math>\\n <jats:tex-math>\\\\sigma \\\\gt 1</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>, is considered in a neighbourhood of the point <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_003.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>z</m:mi>\\n <m:mo>=</m:mo>\\n <m:mn>0</m:mn>\\n </m:math>\\n <jats:tex-math>z=0</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> located either in the complex plane <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_004.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi mathvariant=\\\"double-struck\\\">C</m:mi>\\n </m:math>\\n <jats:tex-math>{\\\\mathbb{C}}</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> if <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_005.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>σ</m:mi>\\n </m:math>\\n <jats:tex-math>\\\\sigma </jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> is a natural number, in a Riemann surface of a rational function if <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_006.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>σ</m:mi>\\n </m:math>\\n <jats:tex-math>\\\\sigma </jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> is a rational number, or in the Riemann surface of logarithmic function if <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_007.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>σ</m:mi>\\n </m:math>\\n <jats:tex-math>\\\\sigma </jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> is an irrational number. It is assumed that <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_008.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>w</m:mi>\\n <m:mo>=</m:mo>\\n <m:mi>w</m:mi>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mi>z</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n </m:math>\\n <jats:tex-math>w=w\\\\left(z)</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>, <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_009.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>a</m:mi>\\n <m:mo>∈</m:mo>\\n <m:mi mathvariant=\\\"double-struck\\\">C</m:mi>\\n <m:mo>⧹</m:mo>\\n <m:mrow>\\n <m:mo>{</m:mo>\\n <m:mrow>\\n <m:mn>0</m:mn>\\n </m:mrow>\\n <m:mo>}</m:mo>\\n </m:mrow>\\n </m:math>\\n <jats:tex-math>a\\\\in {\\\\mathbb{C}}\\\\setminus \\\\left\\\\{0\\\\right\\\\}</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>, and that the function <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_010.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>f</m:mi>\\n </m:math>\\n <jats:tex-math>f</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> is analytic in a neighbourhood of the origin in <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_011.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi mathvariant=\\\"double-struck\\\">C</m:mi>\\n <m:mo>×</m:mo>\\n <m:mi mathvariant=\\\"double-struck\\\">C</m:mi>\\n </m:math>\\n <jats:tex-math>{\\\\mathbb{C}}\\\\times {\\\\mathbb{C}}</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>. Considering <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_012.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>σ</m:mi>\\n </m:math>\\n <jats:tex-math>\\\\sigma </jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0120_eq_013.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>w</m:mi>\\n <m:mo>=</m:mo>\\n <m:mi>w</m:mi>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mi>z</m:mi>\\n \",\"PeriodicalId\":51301,\"journal\":{\"name\":\"Advances in Nonlinear Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Nonlinear Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/anona-2023-0120\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0120","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
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。
Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
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
期刊介绍:
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.