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curl -X POST \
  'http://laksa129.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://link.springer.com/article/10.1007/s11425-017-9303-0\')), getAhrefsUnparsedNoserviceFromURL(\'https://link.springer.com/article/10.1007/s11425-017-9303-0\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0","crawl_time":1747452074,"first_indexed_time":1580698285,"http_code":200,"src_unparsed":"com,springer!link,\/article\/10.1007\/s11425-017-9303-0 s443","src_root_hash":"17645177711233004329","history_drop_reason":null,"meta_title":"Normal forms of linear second order partial differential equations on the plane | Science China Mathematics","meta_descriptions":["The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the resu"],"attrs_boilerpipe_text":"Arnold V I. Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag, 1983 Book \n                    Google Scholar Arnold V I, Ilyashenko Y S. Ordinary differential equations. Encyclopaedia Math Sci, 1988, 1: 1–148 \n                    Google Scholar Arnold V I, Varchenko A N, Gusein-Sade S M. Singularities of Differentiable Mapping, Volume 1. Monographs in Mathematics, vol. 82. Boston: Birkhäuser, 1985 \n                    Google Scholar Bogaevsky I A. Implicit ordinary differential equations: Bifurcations and sharpening of equivalence. Izv Math, 2014, 78: 1063–1078 Article MathSciNet MATH \n                    Google Scholar Bruce J W, Tari F. Generic 1–parameter families of binary differential equations of Morse type. Discrete Contin Dyn Syst, 1997, 3: 79–90 MathSciNet MATH \n                    Google Scholar Bruce J W, Tari F, Fletcher G J. Bifurcations of binary differential equations. Proc Roy Soc Edinburgh Sect A, 2000, 130: 485–506 MathSciNet MATH \n                    Google Scholar Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto. Rend Lombardo, 1932, 65: 889–906 MATH \n                    Google Scholar Dara L. Singularities generiques des equations differentielles multiformes. Bol Soc Bras Mat, 1975, 6: 95–128 Article MathSciNet MATH \n                    Google Scholar Davydov A A. The normal form of a differential equation that is not solved with respect to derivative, in the neighbourhood of its singular point. Funct Anal Appl, 1985, 19: 81–89 Article \n                    Google Scholar Davydov A A. Structural stability of control systems on orientable surfaces. Mat Sb, 1992, 72: 1–28 Article MathSciNet \n                    Google Scholar Davydov A A. Qualitative Theory of Control Systems. Translations of Mathematical Monographs, vol. 141. Providence: Amer Math Soc, 1994 \n                    Google Scholar Davydov A A, Diep L T T. Normal forms for families of linear equations of mixed type near non-resonant folded singular points. Russian Math Surveys, 2010, 65: 984–986 Article MathSciNet MATH \n                    Google Scholar Davydov A A, Diep L T T. Reduction theorem and normal forms of linear second order mixed type PDE families in the plane. TWMS J Pure Appl Math, 2011, 2: 44–53 MathSciNet MATH \n                    Google Scholar Davydov A A, Ishikawa G, Izumiya S, et al. Generic singularities of implicit systems of first order differential equations on the plane. Jpn J Math, 2008, 3: 93–119 Article MathSciNet MATH \n                    Google Scholar Davydov A A, Ortiz-Bobadilla L. Smooth normal forms of folded elementary singular points. J Dyn Control Syst, 1995, 1: 463–483 Article MathSciNet MATH \n                    Google Scholar Davydov A A, Ortiz-Bobadilla L. Normal forms of folded elementary singular points. Russian Math Surveys, 1995, 50: 1260–1261 Article MathSciNet MATH \n                    Google Scholar Davydov A A, Rosales-Gonzales E. Complete classification of generic linear second-order partial differential equations in the plane. Dokl Math, 1996, 350: 151–154 MathSciNet \n                    Google Scholar Davydov A A, Rosales-Gonzales E. Smooth normal forms of folded resonance saddles and nodes and complete classi fication of generic linear second order PDE’s on the plane. In: International Conference on Differential Equation. Singapore: World Scientific, 1998, 59–68 \n                    Google Scholar Grishina Y A, Davydov A A. Structural stability of simplest dynamical inequalities. Proc Steklov Inst Math, 2007, 256: 80–91 Article MathSciNet MATH \n                    Google Scholar Hormander L. On the theory of general partial differential operators. Acta Math, 1955, 94: 161–248 Article MathSciNet MATH \n                    Google Scholar Kasten J A. Solvability of the boundary value problem for a Tricomi type equation in the exterior of a disk. J Math Sci, 2013, 188: 268–272 Article MathSciNet MATH \n                    Google Scholar Kondratiev V A, Landis E M. Qualitative theory of second order linear partial differential equations. 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Lecture Notes on Mixed Type Partial Differential Equations. Singapore: World Scientific, 1990 Book MATH \n                    Google Scholar Smirnov M M. Equations of Mixed Type. Translations of Mathematical Monographs, vol. 51. Providence: Amer Math Soc, 1978 \n                    Google Scholar Sokolov P V. On the paper of A. V. Phadadze and A. A. Šhestakov “on the classification of the singular point of a first order differential equation not solved for the derivative” (in Russian). Mat Sb, 1961, 53: 541–543 MathSciNet \n                    Google Scholar Takens F. Constrained equations: A study of implicit differential equations and their discontinuous solutions. In: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences. Berlin-Heidelberg: Springer, 1976, 143–234 Chapter \n                    Google Scholar Thom R. Sur les equations differentielles multiformes et leurs integrales singulieres. Bol Soc Bras Mat, 1972, 3: 1–11 Article MATH \n                    Google Scholar Tricomi F. Sulle equazioni lineari alle derivate partziali di secondo ordine di tipo misto. Rend Reale Accad Lincei, 1923, 14: 134–247 \n                    Google Scholar","attrs_markdown":"[Skip to main content](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#main)\n\n[![Springer Nature Link](https:\/\/link.springer.com\/oscar-static\/images\/darwin\/header\/img\/logo-springer-nature-link-3149409f62.svg)](https:\/\/link.springer.com\/)\n\n[Account](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0)\n\n[Menu]()\n\n[Find a journal](https:\/\/link.springer.com\/journals\/)   [Publish with us](https:\/\/www.springernature.com\/gp\/authors)   [Track your research](https:\/\/link.springernature.com\/home\/)\n\n[Search]()\n\n[Cart](https:\/\/order.springer.com\/public\/cart)\n## Search\n## Navigation\n- [Find a journal](https:\/\/link.springer.com\/journals\/)\n- [Publish with us](https:\/\/www.springernature.com\/gp\/authors)\n- [Track your research](https:\/\/link.springernature.com\/home\/)\n\n1. [Home](https:\/\/link.springer.com\/)\n2. [Science China Mathematics](https:\/\/link.springer.com\/journal\/11425)\n3. Article\n# Normal forms of linear second order partial differential equations on the plane\n- Reviews\n- [Open access](https:\/\/www.springernature.com\/gp\/open-research\/about\/the-fundamentals-of-open-access-and-open-research)\n- Published:\n  13 September 2018\n\n- Volume 61, pages 1947–1962, (2018)\n- [Cite this article](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#citeas)\n\n[Download PDF](https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11425-017-9303-0.pdf)\n\nYou have full access to this [open access](https:\/\/www.springernature.com\/gp\/open-research\/about\/the-fundamentals-of-open-access-and-open-research) article\n\n[![](https:\/\/media.springernature.com\/w72\/springer-static\/cover-hires\/journal\/11425?as=webp) Science China Mathematics](https:\/\/link.springer.com\/journal\/11425)\n\n[Aims and scope](https:\/\/link.springer.com\/journal\/11425\/aims-and-scope)\n\n[Submit manuscript](https:\/\/mc03.manuscriptcentral.com\/scmath)\n\nNormal forms of linear second order partial differential equations on the plane\n\n[Download PDF](https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11425-017-9303-0.pdf)\n\n- [Alexey Davydov](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#auth-Alexey-Davydov-Aff1-Aff2-Aff3)\n  [1](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Aff1),[2](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Aff2),[3](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Aff3)\n\n## Abstract\nThe paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recent results in the theory, is included in the paper in the parametric form together with proof. There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.\n\n## Article PDF\n[Download](https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11425-017-9303-0.pdf) to read the full article text\n\n### Similar content being viewed by others\n![](https:\/\/media.springernature.com\/w215h120\/springer-static\/image\/art%3Aplaceholder%2Fimages\/placeholder-figure-springernature.png)\n\n### [Geometric Methods in Partial Differential Equations](https:\/\/link.springer.com\/10.1007\/s00032-021-00336-9)\nArticle 22 November 2021\n\n![](https:\/\/media.springernature.com\/w215h120\/springer-static\/image\/art%3Aplaceholder%2Fimages\/placeholder-figure-springernature.png)\n\n### [Normal Forms of Planar Polynomial Differential Systems](https:\/\/link.springer.com\/10.1007\/s12346-018-0273-4)\nArticle 12 February 2018\n\n![](https:\/\/media.springernature.com\/w215h120\/springer-static\/image\/art%3A10.1007%2Fs12346-015-0174-8\/MediaObjects\/12346_2015_174_Fig1_HTML.gif)\n\n### [Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions](https:\/\/link.springer.com\/10.1007\/s12346-015-0174-8)\nArticle 09 November 2015\n\n### Explore related subjects\nDiscover the latest articles and news from researchers in related subjects, suggested using machine learning.\n\n- [Differential Equations](https:\/\/link.springer.com\/subjects\/differential-equations)\n- [Differential Geometry](https:\/\/link.springer.com\/subjects\/differential-geometry)\n- [Ordinary Differential Equations](https:\/\/link.springer.com\/subjects\/ordinary-differential-equations)\n- [Partial Differential Equations](https:\/\/link.springer.com\/subjects\/partial-differential-equations)\n- [Functional Analysis](https:\/\/link.springer.com\/subjects\/functional-analysis)\n- [Partial Differential Equations on Manifolds](https:\/\/link.springer.com\/subjects\/partial-differential-equations-on-manifolds)\n\n[Use our pre-submission checklist](https:\/\/beta.springernature.com\/pre-submission?journalId=11425)\n\nAvoid common mistakes on your manuscript.\n\n## References\n1. 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Department of Theory of Dynamical Systems, Lomonosov Moscow State University, Moscow, 119991, Russia\n   Alexey Davydov\n3. International Institute for Applied Systems Analysis, Laxenburg, 2361, Austria\n   Alexey Davydov\n\nAuthors\n\n1. 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article\nDavydov, A. Normal forms of linear second order partial differential equations on the plane. *Sci. China Math.* **61**, 1947–1962 (2018). https:\/\/doi.org\/10.1007\/s11425-017-9303-0\n\n[Download citation](https:\/\/citation-needed.springer.com\/v2\/references\/10.1007\/s11425-017-9303-0?format=refman&flavour=citation)\n\n- Received: 03 October 2017\n- Accepted: 18 April 2018\n- Published: 13 September 2018\n- Issue Date: November 2018\n- DOI: https:\/\/doi.org\/10.1007\/s11425-017-9303-0\n\n### Share this article\nAnyone you share the following link with will be able to read this content:\n\nGet shareable link\n\nSorry, a shareable link is not currently available for this article.\n\nCopy to clipboard\n\nProvided by the Springer Nature SharedIt content-sharing initiative\n### Keywords\n- [normal form](https:\/\/link.springer.com\/search?query=normal%20form&facet-discipline=\"Mathematics\")\n- [mixed type partial differential equation](https:\/\/link.springer.com\/search?query=mixed%20type%20partial%20differential%20equation&facet-discipline=\"Mathematics\")\n- [main symbol](https:\/\/link.springer.com\/search?query=main%20symbol&facet-discipline=\"Mathematics\")\n### MSC(2010)\n- [34A09](https:\/\/link.springer.com\/search?query=34A09&facet-discipline=\"Mathematics\")\n- [34C20](https:\/\/link.springer.com\/search?query=34C20&facet-discipline=\"Mathematics\")\n\n[Use our pre-submission checklist](https:\/\/beta.springernature.com\/pre-submission?journalId=11425)\n\nAvoid common mistakes on your manuscript.\n\n- Sections\n- References\n\n- [Abstract](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Abs1)\n- [Article PDF](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#preview)\n- [References](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Bib1)\n- [Acknowledgements](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#Ack1)\n- [Author information](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#author-information)\n- [Rights and permissions](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#rightslink)\n- [About this article](https:\/\/link.springer.com\/article\/10.1007\/s11425-017-9303-0#article-info)\n\nAdvertisement\n\n1. 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Boilerpipe Text	Arnold V I. Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag, 1983 Book Google Scholar Arnold V I, Ilyashenko Y S. Ordinary differential equations. Encyclopaedia Math Sci, 1988, 1: 1–148 Google Scholar Arnold V I, Varchenko A N, Gusein-Sade S M. Singularities of Differentiable Mapping, Volume 1. Monographs in Mathematics, vol. 82. Boston: Birkhäuser, 1985 Google Scholar Bogaevsky I A. Implicit ordinary differential equations: Bifurcations and sharpening of equivalence. Izv Math, 2014, 78: 1063–1078 Article MathSciNet MATH Google Scholar Bruce J W, Tari F. Generic 1–parameter families of binary differential equations of Morse type. Discrete Contin Dyn Syst, 1997, 3: 79–90 MathSciNet MATH Google Scholar Bruce J W, Tari F, Fletcher G J. Bifurcations of binary differential equations. Proc Roy Soc Edinburgh Sect A, 2000, 130: 485–506 MathSciNet MATH Google Scholar Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto. Rend Lombardo, 1932, 65: 889–906 MATH Google Scholar Dara L. Singularities generiques des equations differentielles multiformes. Bol Soc Bras Mat, 1975, 6: 95–128 Article MathSciNet MATH Google Scholar Davydov A A. The normal form of a differential equation that is not solved with respect to derivative, in the neighbourhood of its singular point. Funct Anal Appl, 1985, 19: 81–89 Article Google Scholar Davydov A A. Structural stability of control systems on orientable surfaces. Mat Sb, 1992, 72: 1–28 Article MathSciNet Google Scholar Davydov A A. Qualitative Theory of Control Systems. Translations of Mathematical Monographs, vol. 141. Providence: Amer Math Soc, 1994 Google Scholar Davydov A A, Diep L T T. Normal forms for families of linear equations of mixed type near non-resonant folded singular points. Russian Math Surveys, 2010, 65: 984–986 Article MathSciNet MATH Google Scholar Davydov A A, Diep L T T. Reduction theorem and normal forms of linear second order mixed type PDE families in the plane. TWMS J Pure Appl Math, 2011, 2: 44–53 MathSciNet MATH Google Scholar Davydov A A, Ishikawa G, Izumiya S, et al. Generic singularities of implicit systems of first order differential equations on the plane. Jpn J Math, 2008, 3: 93–119 Article MathSciNet MATH Google Scholar Davydov A A, Ortiz-Bobadilla L. Smooth normal forms of folded elementary singular points. J Dyn Control Syst, 1995, 1: 463–483 Article MathSciNet MATH Google Scholar Davydov A A, Ortiz-Bobadilla L. Normal forms of folded elementary singular points. Russian Math Surveys, 1995, 50: 1260–1261 Article MathSciNet MATH Google Scholar Davydov A A, Rosales-Gonzales E. Complete classification of generic linear second-order partial differential equations in the plane. Dokl Math, 1996, 350: 151–154 MathSciNet Google Scholar Davydov A A, Rosales-Gonzales E. Smooth normal forms of folded resonance saddles and nodes and complete classi fication of generic linear second order PDE’s on the plane. In: International Conference on Differential Equation. Singapore: World Scientific, 1998, 59–68 Google Scholar Grishina Y A, Davydov A A. Structural stability of simplest dynamical inequalities. Proc Steklov Inst Math, 2007, 256: 80–91 Article MathSciNet MATH Google Scholar Hormander L. On the theory of general partial differential operators. Acta Math, 1955, 94: 161–248 Article MathSciNet MATH Google Scholar Kasten J A. Solvability of the boundary value problem for a Tricomi type equation in the exterior of a disk. J Math Sci, 2013, 188: 268–272 Article MathSciNet MATH Google Scholar Kondratiev V A, Landis E M. Qualitative theory of second order linear partial differential equations. 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Article # Normal forms of linear second order partial differential equations on the plane - Reviews - [Open access](https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research) - Published: 13 September 2018 - Volume 61, pages 1947–1962, (2018) - [Cite this article](https://link.springer.com/article/10.1007/s11425-017-9303-0#citeas) [Download PDF](https://link.springer.com/content/pdf/10.1007/s11425-017-9303-0.pdf) You have full access to this [open access](https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research) article [![](https://media.springernature.com/w72/springer-static/cover-hires/journal/11425?as=webp) Science China Mathematics](https://link.springer.com/journal/11425) [Aims and scope](https://link.springer.com/journal/11425/aims-and-scope) [Submit manuscript](https://mc03.manuscriptcentral.com/scmath) Normal forms of linear second order partial differential equations on the plane [Download PDF](https://link.springer.com/content/pdf/10.1007/s11425-017-9303-0.pdf) - [Alexey Davydov](https://link.springer.com/article/10.1007/s11425-017-9303-0#auth-Alexey-Davydov-Aff1-Aff2-Aff3) [1](https://link.springer.com/article/10.1007/s11425-017-9303-0#Aff1),[2](https://link.springer.com/article/10.1007/s11425-017-9303-0#Aff2),[3](https://link.springer.com/article/10.1007/s11425-017-9303-0#Aff3) ## Abstract The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recent results in the theory, is included in the paper in the parametric form together with proof. There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory. ## Article PDF [Download](https://link.springer.com/content/pdf/10.1007/s11425-017-9303-0.pdf) to read the full article text ### Similar content being viewed by others ![](https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png) ### [Geometric Methods in Partial Differential Equations](https://link.springer.com/10.1007/s00032-021-00336-9) Article 22 November 2021 ![](https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png) ### [Normal Forms of Planar Polynomial Differential Systems](https://link.springer.com/10.1007/s12346-018-0273-4) Article 12 February 2018 ![](https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs12346-015-0174-8/MediaObjects/12346_2015_174_Fig1_HTML.gif) ### [Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions](https://link.springer.com/10.1007/s12346-015-0174-8) Article 09 November 2015 ### Explore related subjects Discover the latest articles and news from researchers in related subjects, suggested using machine learning. - [Differential Equations](https://link.springer.com/subjects/differential-equations) - [Differential Geometry](https://link.springer.com/subjects/differential-geometry) - [Ordinary Differential Equations](https://link.springer.com/subjects/ordinary-differential-equations) - [Partial Differential Equations](https://link.springer.com/subjects/partial-differential-equations) - [Functional Analysis](https://link.springer.com/subjects/functional-analysis) - [Partial Differential Equations on Manifolds](https://link.springer.com/subjects/partial-differential-equations-on-manifolds) [Use our pre-submission checklist](https://beta.springernature.com/pre-submission?journalId=11425) Avoid common mistakes on your manuscript. ## References 1. 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