複素微分方程式のソースを表示

{{Differential equations}}
'''複素微分方程式'''（ふくそびぶんほうていしき、{{lang-en-short|Complex differential equations}}）は、[[複素関数]]を厳密解としてもつ[[微分方程式]]の総称であり、その解析には[[解析接続]]や[[モノドロミー行列]]をはじめとした[[複素解析]]の道具が用いられる<ref name="af">Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. [[:en:Cambridge University Press]].</ref><ref name="huku"/><ref name="taka"/><ref name="kim"/>。
==主な複素微分方程式==
===主な複素常微分方程式===
{{seealso|常微分方程式}}
{{div col|rules=yes}}
* [[超幾何微分方程式]]<ref name="taka"/><ref name="g2p">Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions. [[:en:Springer Science & Business Media]].</ref><ref>原岡喜重. (2002). 超幾何関数. [[朝倉書店]].</ref><ref>木村弘信: 超幾何関数入門——特殊関数への統一的視点からのアプローチ——, [[サイエンス社]], 2007 年.</ref>
* [[パンルヴェ方程式]]<ref name="g2p"/><ref>Bruno, A. D., & Batkhin, A. B. (Eds.). (2012). Painlevé Equations and Related Topics. de Gruyter.</ref><ref>Noumi, M. (2004). Painlevé equations through symmetry. [[:en:Springer Science & Business Media]].</ref><ref>Bobenko, A. I., Berlin, A. I. B. T., & Eitner, U. (2000). Painlevé equations in the differential geometry of surfaces. [[:en:Springer Science & Business Media]].</ref><ref name="sophia">岡本和夫. (1985). パンルヴェ方程式序説. [[上智大学]]数学講究録, 19.</ref><ref name="oka">岡本和夫. (2009). パンルヴェ方程式. [[岩波書店]].</ref><ref>野海正俊. (2000). パンルヴェ方程式-対称性からの入門. すうがくの風景 4. [[朝倉書店]].</ref>
* [[ホイン函数#ホインの方程式]]<ref>Maier, R. (2007). The 192 solutions of the Heun equation. [[:en:Mathematics of Computation]], 76(258), 811-843.</ref><ref>Maier, R. S. (2005). On reducing the Heun equation to the hypergeometric equation. Journal of Differential Equations, 213(1), 171-203.</ref>
* [[リッカチの微分方程式]]<ref>Bittanti, S., Laub, A. J., & Willems, J. C. (Eds.). (2012). The Riccati Equation. Springer Science & Business Media.</ref><ref> リッカチのひ・み・つ リッカチ方程式の解けるしくみ. 井ノ口 順一; [[日本評論社]].</ref>
{{div col end}}

===主な複素偏微分方程式===
{{seealso|偏微分方程式}}
{{div col|rules=yes}}
* [[非線形シュレディンガー方程式]]<ref>Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory (2001), Zhidkov, Peter E., Springer.</ref><ref>The Nonlinear Schrödinger Equation (1999) -Self-Focusing and Wave Collapse- , Sulem, Catherine, Sulem, Pierre-Louis, Springer.</ref><ref>The Nonlinear Schrödinger Equation -Singular Solutions and Optical Collapse- (2015), Gadi Fibich, Springer.</ref>
* 複素[[KdV方程式]]<ref>Birnir, B. (1987). An example of blow-up, for the complex KdV equation and existence beyond the blow-up. SIAM Journal on Applied Mathematics, 47(4), 710-725.</ref><ref>Zhang, Y., Lv, Y. N., Ye, L. Y., & Zhao, H. Q. (2007). The exact solutions to the complex KdV equation. Physics Letters A, 367(6), 465-472.</ref><ref>An, H. L., & Chen, Y. (2008). Numerical complexiton solutions for the complex KdV equation by the homotopy perturbation method. Applied Mathematics and Computation, 203(1), 125-133.</ref><ref>Yuan, J. M., & Wu, J. (2005). The complex KdV equation with or without dissipation. Discrete Contin. Dyn. Syst. Ser. B, 5, 489-512.</ref>
* 複素[[mKdV方程式]]<ref>Ma, L. Y., Shen, S. F., & Zhu, Z. N. (2016). Integrable nonlocal complex mKdV equation: soliton solution and gauge equivalence. arXiv preprint arXiv:1612.06723.</ref><ref>Qi-Lao, Z., & Zhi-Bin, L. (2008). Darboux transformation and multi-solitons for complex mKdV equation. Chinese Physics Letters, 25(1), 8.</ref><ref>Anco, S. C., Ngatat, N. T., & Willoughby, M. (2011). Interaction properties of complex modified Korteweg–de Vries (mKdV) solitons. Physica D: Nonlinear Phenomena, 240(17), 1378-1394.</ref><ref>He, J., Wang, L., Li, L., Porsezian, K., & Erdélyi, R. (2014). Few-cycle optical rogue waves: complex modified Korteweg–de Vries equation. Physical Review E, 89(6), 062917.</ref>
* 複素[[バーガース方程式]]<ref>Kenyon, R., & Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Mathematica, 199(2), 263-302.</ref><ref>Liu, T. P., & Zumbrun, K. (1995). Nonlinear stability of an undercompressive shock for complex Burgers equation. Communications in mathematical physics, 168(1), 163-186.</ref><ref>Jia-Qi, M., & Xian-Feng, C. (2010). Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation. Chinese Physics B, 19(10), 100203.</ref><ref>Neuberger, H. (2008). Complex Burgers' equation in 2D SU (N) YM. Physics Letters B, 670(3), 235-240.</ref><ref>Senouf, D., Caflisch, R., & Ercolani, N. (1996). Pole dynamics and oscillations for the complex Burgers equation in the small-dispersion limit. Nonlinearity, 9(6), 1671.</ref>
* 複素短パルス方程式 ({{lang-en-short|complex short pulse equation}})<ref>Shen, S., Feng, B. F., & Ohta, Y. (2016). From the real and complex coupled dispersionless equations to the real and complex short pulse equations. Studies in Applied Mathematics, 136(1), 64-88.</ref>
* 複素サイン・ゴルドン方程式 ({{lang-en-short|complex sine-Gordon equation}})<ref>Park, Q. H., & Shin, H. J. (1995). Duality in complex sine-Gordon theory. Physics Letters B, 359(1-2), 125-132.
</ref><ref>Aratyn, H., Ferreira, L. A., Gomes, J. F., & Zimerman, A. H. (2000). The complex sine-Gordon equation as a symmetry flow of the AKNS hierarchy. Journal of Physics A: Mathematical and General, 33(35), L331.</ref><ref>Barashenkov, I. V., & Pelinovsky, D. E. (1998). Exact vortex solutions of the complex sine-Gordon theory on the plane. Physics Letters B, 436(1-2), 117-124.</ref><ref>Park, Q. H., & Shin, H. J. (1999). Complex sine-Gordon equation in coherent optical pulse propagation. arXiv preprint solv-int/9904007.</ref><ref>Sergyeyev, A., & Demskoi, D. (2007). Sasa-Satsuma (complex modified Korteweg–de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more. Journal of mathematical physics, 48(4), 042702.</ref><ref>Getmanov, B. S. (1981). Integrable two-dimensional Lorentz-invariant nonlinear model of a complex scalar field (complex sine-Gordon II). Theoretical and Mathematical Physics, 48(1), 572-579.</ref>
* 複素 [[ギンツブルグ-ランダウ理論|Ginzburg-Landau 方程式]]<ref>Aranson, I. S., & Kramer, L. (2002). The world of the complex Ginzburg-Landau equation. Reviews of Modern Physics, 74(1), 99.</ref><ref>Akhmediev, N. N., Ankiewicz, A., & Soto-Crespo, J. M. (1997). Multisoliton solutions of the complex Ginzburg-Landau equation. Physical review letters, 79(21), 4047.</ref><ref>Shraiman, B. I., Pumir, A., van Saarloos, W., [[ピエール・ホーエンバーグ|Hohenberg, P. C.]], Chaté, H., & Holen, M. (1992). Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. Physica D: Nonlinear Phenomena, 57(3-4), 241-248.</ref><ref>Van Saarloos, W., & [[ピエール・ホーエンバーグ|Hohenberg, P. C.]] (1990). Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Physical review letters, 64(7), 749.</ref><ref>Chate, H. (1994). Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation. Nonlinearity, 7(1), 185.</ref><ref>Doering, C. R., Gibbon, J. D., Holm, D. D., & Nicolaenko, B. (1988). Low-dimensional behaviour in the complex Ginzburg-Landau equation. Nonlinearity, 1(2), 279.</ref><ref>Hakim, V., & Rappel, W. J. (1992). Dynamics of the globally coupled complex Ginzburg-Landau equation. Physical Review A, 46(12), R7347.</ref><ref>Chaté, H., & Manneville, P. (1996). Phase diagram of the two-dimensional complex Ginzburg-Landau equation. Physica A: Statistical Mechanics and its Applications, 224(1-2), 348-368.</ref><ref>Battogtokh, D., & Mikhailov, A. (1996). Controlling turbulence in the complex Ginzburg-Landau equation. Physica D: Nonlinear Phenomena, 90(1-2), 84-95.</ref>
{{div col end}}

==研究者==
===日本===
{{div col|rules=yes}}
* [[福原満洲雄]]<ref name="huku">[[福原満洲雄]]「常微分方程式 第2版」岩波全書.</ref><ref>M. Hukuhara, Quelques remarques sur le mémoire de P. Painlevé: Sur les &eaute;quations différentielles dont l'intégrale générale est uniforme, Publ. Res. Inst. Math. Sci. Ser. A, 3 (1967), 139-150.</ref><ref>M. Hukuhara, T. Kimura, T. Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe, Publ. Math. Soc. Japan, 7. [[日本数学会|The Mathematical Society of Japan]], Tokyo 1961.</ref><ref>[[福原満洲雄]]. (1982). 常微分方程式の 50 年, II. 数学, 34(3), 262-269.</ref><ref>[[福原満洲雄]], & 斎藤ユリ子. (1971). 大域的な理論による特殊関数の取扱い (解析的常微分方程式の大域的研究).</ref>
* 大島利雄<ref>大島利雄 述, & 廣惠一希 記. (2011). [[特殊関数]]と代数的線型常微分方程式. Lecture Notes in Mathematical Sciences, 11.</ref><ref>大島利雄. [[リーマン球面|Riemann 球面]]上の複素常微分方程式と多変数超幾何函数. 第15回 岡シンポジウム (2015 年) の講義録として出版予定, [[奈良女子大学]].</ref><ref>大島利雄. (1972). 定数係数線型偏微分方程式系の解の存在について (超函数と微分方程式).</ref><ref>大島利雄. (2017). KZ 型超幾何系の変換と解析 (表現論と非可換[[調和解析]]をめぐる諸問題).</ref>
* 岡本和夫<ref name="sophia"/><ref name="oka"/><ref>岡本和夫. (1986). 日仏セミナー ‘複素領域における微分方程式論’. 数学, 38(3), 277-282.</ref><ref>岡本和夫. (1977). Painleve の方程式によって定義される葉層構造について (微分方程式の幾何学的方法).</ref><ref>岡本和夫. (1974). 非線型常微分方程式の解の幾何的性質についての一考察 (常微分方程式の解析的理論: 解の接続).</ref><ref>岡本和夫. (1980). Painlevé の方程式. 数学, 32(1), 30-43.</ref>
* 高野恭一<ref name="taka">常微分方程式, [[朝倉書店]], 高野恭一.</ref><ref>Takano, K., Reduction for Painlevé equations at the fixed singular points of the first kind, Funkcial. Ekvac., 29(1986), 99-119.</ref><ref>Takano, K., Reduction for Painlevé equations at the fixed singular points of the second kind, J. Math. Soc. Japan, 42(1990), 423-443.</ref><ref>Kimura, H.,Matumiya, A. and Takano, K., A normal form of Hamiltonian systems of several time variables with a regular singularity, J. Differential Equations, 127(1996), 337-364.</ref><ref>Shioda, T. and Takano, K., On some Hamiltonian structures of Painlevé systems, I, Funkcial. Ekvac., 40(1997), 271-291.</ref><ref>Matano, T., Matumiya, A. and Takano, K., On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, 51(1999), 843-866.</ref><ref>Takano, K., Defining manifolds for Painlevé equations. "Toward the exact WKB analysis of differential equations, linear or non-linear" (Eds. C.J. Howls, T. Kawai, and Y. Takei), 261-269, Kyoto Univ. Press, Kyoto, 2000.</ref><ref>Takano, K., Confluences of defining manifolds of Painlevé systems, Tohoku Math. J., 53(2001), 319-335.</ref><ref>Noumi, M., Takano, K. and Yamada, Y., Bäcklund transformations and the manifolds of Painlevé systems, Funkcial. Ekvac., 45(2002), 237-258.</ref><ref>Suzuki, M., Tahara, N. and Takano, K., Hierarchy of Bäcklund transformation groups of the Painlevé systems, J. Math. Soc. Japan, 56(2004), 1221-1232.</ref><ref>Kimura, H. and Takano, K., On confluences of general hypergeometric systems, Tohoku Math. J., 58(2006), 1-31.</ref>
* [[木村俊房]]<ref name="kim">[[木村俊房]]「常微分方程式II」基礎数学講座.</ref>
* [[神保道夫]]<ref>Jimbo, M. (1982). Monodromy problem and the boundary condition for some Painlevé equations. Publications of the [[京都大学数理解析研究所|Research Institute for Mathematical Sciences]], 18(3), 1137-1161.</ref><ref>Jimbo, M., Miwa, T., Môri, Y., & Sato, M. (1980). Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D: Nonlinear Phenomena, 1(1), 80-158.</ref><ref>Jimbo, M., Miwa, T., & Ueno, K. (1981). Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and <math>\tau</math>-function. Physica D: Nonlinear Phenomena, 2(2), 306-352.</ref><ref>Jimbo, M., & Miwa, T. (1981). Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D: Nonlinear Phenomena, 2(3), 407-448.</ref><ref>Jimbo, M., & Miwa, T. (1981). Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III. Physica D: Nonlinear Phenomena, 4(1), 26-46.</ref>
{{div col end}}

===海外===
{{div col|rules=yes}}
* [[ポール・パンルヴェ]]
* [[:en:Lazarus Fuchs]]
* [[ダフィット・ヒルベルト]]
* [[ベルンハルト・リーマン]]
* [[:en:Alexander Its]]<ref name="its">Fokas, A. S., Its, A. R., Novokshenov, V. Y., Kapaev, A. A., Kapaev, A. I., & Novokshenov, V. Y. (2006). Painlevé transcendents: the Riemann-Hilbert approach. [[アメリカ数学会|American Mathematical Society]].</ref><ref>Its, A. R., & Novokshenov, V. Y. (2006). The isomonodromic deformation method in the theory of Painlevé equations. Springer.</ref>
* [[:en:Mark J. Ablowitz]]<ref name="af"/><ref>Fokas, A. S., & Ablowitz, M. J. (1981). Linearization of the Korteweg—de Vries and Painlevé II Equations. Physical Review Letters, 47(16), 1096.</ref><ref>Fokas, A. S., & Ablowitz, M. J. (1982). On a unified approach to transformations and elementary solutions of Painlevé equations. Journal of Mathematical Physics, 23(11), 2033-2042.</ref><ref>Fokas, A. S., Mugan, U., & Ablowitz, M. J. (1988). A method of linearization for Painlevé equations: Painlevé IV, V. Physica D: Nonlinear Phenomena, 30(3), 247-283.</ref><ref>Fokas, A. S., & Ablowitz, M. J. (1983). On the initial value problem of the second Painlevé transcendent. Communications in mathematical physics, 91(3), 381-403.</ref>
{{div col end}}

==関連項目==
* [[フックス型微分方程式]]<ref name="taka"/>
* [[コーシー＝コワレフスカヤの定理]]
* [[クニーズニク・ザモロドチコフ方程式]]<ref>Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, 58, [[アメリカ数学会|American Mathematical Society]], {{ISBN2|0821804960}}</ref>
* [[:en:Riemann–Hilbert problem]]<ref name="af"/><ref name="taka"/><ref name="its"/><ref>Trogdon, T., & Olver, S. (2015). Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions. [[SIAM (学会)|SIAM]].</ref>
* [[:en:Riemann–Hilbert correspondence]]

==出典==
{{脚注ヘルプ}}
{{reflist|2}}
==参考文献==
{{参照方法|date=2019年10月}}
* Einar Hille (1976). Ordinary Differential Equations in the Complex Domain. Wiley. {{ISBN2|978-0-471-39964-3}}., reprinted by Dover, 1997.
* E. Ince (1926). Ordinary Differential Equations. Dover., reprinted by Dover, 2003.
* Gromak, Laine, Shimomura (2002). Painlevé Differential Equations in the Complex Plane. de Gruyter. {{ISBN2|978-3-11-017379-6}}.
* Ilpo Laine (1992). [[ネヴァンリンナ理論|Nevanlinna Theory]] and Complex Differential Equations. de Gruyter. {{ISBN2|978-3-11-013422-3}}.
* Eremenko, A. (1982). "Meromorphic solutions of algebraic differential equations". Russian Mathematical Surveys. 37 (4): 61–94. CiteSeerX 10.1.1.139.8499. doi:10.1070/RM1982v037n04ABEH003967.
* So-Chin Chen; Mei-Chi Shaw (2002). Partial Differential Equations in [[多変数複素関数|Several Complex Variables]]. [[アメリカ数学会|American Mathematical Society]]. {{ISBN2|978-0-8218-2961-5}}.

{{数学}}
{{math-stub}}

[[Category:複素解析]]
[[Category:微分方程式]]
[[Category:常微分方程式]]
[[Category:偏微分方程式]]
[[Category:数学に関する記事]]
{{デフォルトソート:ふくそひふんほうていしき}}