複素微分方程式

複素微分方程式（ふくそびぶんほうていしき、英: 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^[1]^[2]^[3]^[4]。

主な複素微分方程式

主な複素常微分方程式

「常微分方程式」も参照

主な複素偏微分方程式

「偏微分方程式」も参照

非線形シュレディンガー方程式^[18]^[19]^[20]
複素KdV方程式^[21]^[22]^[23]^[24]
複素mKdV方程式^[25]^[26]^[27]^[28]
複素バーガース方程式^[29]^[30]^[31]^[32]^[33]
複素短パルス方程式 (英: complex short pulse equation)^[34]
複素サイン・ゴルドン方程式 (英: complex sine-Gordon equation)^[35]^[36]^[37]^[38]^[39]^[40]
複素 Ginzburg-Landau 方程式^[41]^[42]^[43]^[44]^[45]^[46]^[47]^[48]^[49]

研究者

日本

福原満洲雄^[2]^[50]^[51]^[52]^[53]
大島利雄^[54]^[55]^[56]^[57]
岡本和夫^[11]^[12]^[58]^[59]^[60]^[61]
高野恭一^[3]^[62]^[63]^[64]^[65]^[66]^[67]^[68]^[69]^[70]^[71]
木村俊房^[4]
神保道夫^[72]^[73]^[74]^[75]^[76]

海外

出典

[脚注の使い方]

^ ^a ^b ^c Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. en:Cambridge University Press.
^ ^a ^b 福原満洲雄「常微分方程式第2版」岩波全書.
^ ^a ^b ^c ^d ^e 常微分方程式, 朝倉書店, 高野恭一.
^ ^a ^b 木村俊房「常微分方程式II」基礎数学講座.
^ ^a ^b Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions. en:Springer Science & Business Media.
^ 原岡喜重. (2002). 超幾何関数. 朝倉書店.
^ 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.
^ Bruno, A. D., & Batkhin, A. B. (Eds.). (2012). Painlevé Equations and Related Topics. de Gruyter.
^ Noumi, M. (2004). Painlevé equations through symmetry. en:Springer Science & Business Media.
^ Bobenko, A. I., Berlin, A. I. B. T., & Eitner, U. (2000). Painlevé equations in the differential geometry of surfaces. en:Springer Science & Business Media.
^ ^a ^b 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.
^ ^a ^b 岡本和夫. (2009). パンルヴェ方程式. 岩波書店.
^ 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.
^ Maier, R. (2007). The 192 solutions of the Heun equation. en:Mathematics of Computation, 76(258), 811-843.
^ Maier, R. S. (2005). On reducing the Heun equation to the hypergeometric equation. Journal of Differential Equations, 213(1), 171-203.
^ Bittanti, S., Laub, A. J., & Willems, J. C. (Eds.). (2012). The Riccati Equation. Springer Science & Business Media.
^ 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.
^ Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory (2001), Zhidkov, Peter E., Springer.
^ The Nonlinear Schrödinger Equation (1999) -Self-Focusing and Wave Collapse- , Sulem, Catherine, Sulem, Pierre-Louis, Springer.
^ The Nonlinear Schrödinger Equation -Singular Solutions and Optical Collapse- (2015), Gadi Fibich, Springer.
^ 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.
^ 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.
^ 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.
^ Yuan, J. M., & Wu, J. (2005). The complex KdV equation with or without dissipation. Discrete Contin. Dyn. Syst. Ser. B, 5, 489-512.
^ 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.
^ Qi-Lao, Z., & Zhi-Bin, L. (2008). Darboux transformation and multi-solitons for complex mKdV equation. Chinese Physics Letters, 25(1), 8.
^ 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.
^ 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.
^ Kenyon, R., & Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Mathematica, 199(2), 263-302.
^ Liu, T. P., & Zumbrun, K. (1995). Nonlinear stability of an undercompressive shock for complex Burgers equation. Communications in mathematical physics, 168(1), 163-186.
^ 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.
^ Neuberger, H. (2008). Complex Burgers' equation in 2D SU (N) YM. Physics Letters B, 670(3), 235-240.
^ 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.
^ 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.
^ Park, Q. H., & Shin, H. J. (1995). Duality in complex sine-Gordon theory. Physics Letters B, 359(1-2), 125-132.
^ 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.
^ 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.
^ Park, Q. H., & Shin, H. J. (1999). Complex sine-Gordon equation in coherent optical pulse propagation. arXiv preprint solv-int/9904007.
^ 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.
^ 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.
^ Aranson, I. S., & Kramer, L. (2002). The world of the complex Ginzburg-Landau equation. Reviews of Modern Physics, 74(1), 99.
^ Akhmediev, N. N., Ankiewicz, A., & Soto-Crespo, J. M. (1997). Multisoliton solutions of the complex Ginzburg-Landau equation. Physical review letters, 79(21), 4047.
^ Shraiman, B. I., Pumir, A., van Saarloos, W., 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.
^ Van Saarloos, W., & Hohenberg, P. C. (1990). Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Physical review letters, 64(7), 749.
^ Chate, H. (1994). Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation. Nonlinearity, 7(1), 185.
^ 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.
^ Hakim, V., & Rappel, W. J. (1992). Dynamics of the globally coupled complex Ginzburg-Landau equation. Physical Review A, 46(12), R7347.
^ 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.
^ Battogtokh, D., & Mikhailov, A. (1996). Controlling turbulence in the complex Ginzburg-Landau equation. Physica D: Nonlinear Phenomena, 90(1-2), 84-95.
^ 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.
^ M. Hukuhara, T. Kimura, T. Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe, Publ. Math. Soc. Japan, 7. The Mathematical Society of Japan, Tokyo 1961.
^ 福原満洲雄. (1982). 常微分方程式の 50 年, II. 数学, 34(3), 262-269.
^ 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).
^ 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.
^ 大島利雄. 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.
^ 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).
^ 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).
^ 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.
^ 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).
^ 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).
^ 岡本和夫. (1980). Painlevé の方程式. 数学, 32(1), 30-43.
^ Takano, K., Reduction for Painlevé equations at the fixed singular points of the first kind, Funkcial. Ekvac., 29(1986), 99-119.
^ Takano, K., Reduction for Painlevé equations at the fixed singular points of the second kind, J. Math. Soc. Japan, 42(1990), 423-443.
^ 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.
^ Shioda, T. and Takano, K., On some Hamiltonian structures of Painlevé systems, I, Funkcial. Ekvac., 40(1997), 271-291.
^ Matano, T., Matumiya, A. and Takano, K., On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, 51(1999), 843-866.
^ 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.
^ Takano, K., Confluences of defining manifolds of Painlevé systems, Tohoku Math. J., 53(2001), 319-335.
^ Noumi, M., Takano, K. and Yamada, Y., Bäcklund transformations and the manifolds of Painlevé systems, Funkcial. Ekvac., 45(2002), 237-258.
^ Suzuki, M., Tahara, N. and Takano, K., Hierarchy of Bäcklund transformation groups of the Painlevé systems, J. Math. Soc. Japan, 56(2004), 1221-1232.
^ Kimura, H. and Takano, K., On confluences of general hypergeometric systems, Tohoku Math. J., 58(2006), 1-31.
^ Jimbo, M. (1982). Monodromy problem and the boundary condition for some Painlevé equations. Publications of the Research Institute for Mathematical Sciences, 18(3), 1137-1161.
^ 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.
^ Jimbo, M., Miwa, T., & Ueno, K. (1981). Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and $\tau$ -function. Physica D: Nonlinear Phenomena, 2(2), 306-352.
^ 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.
^ 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.
^ ^a ^b 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. American Mathematical Society.
^ Its, A. R., & Novokshenov, V. Y. (2006). The isomonodromic deformation method in the theory of Painlevé equations. Springer.
^ Fokas, A. S., & Ablowitz, M. J. (1981). Linearization of the Korteweg—de Vries and Painlevé II Equations. Physical Review Letters, 47(16), 1096.
^ 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.
^ 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.
^ 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.
^ Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, 58, American Mathematical Society, ISBN 0821804960
^ Trogdon, T., & Olver, S. (2015). Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions. SIAM.

参考文献

出典は列挙するだけでなく、脚注などを用いてどの記述の情報源であるかを明記してください。記事の信頼性向上にご協力をお願いいたします。（2019年10月）

Einar Hille (1976). Ordinary Differential Equations in the Complex Domain. Wiley. ISBN 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. ISBN 978-3-11-017379-6.
Ilpo Laine (1992). Nevanlinna Theory and Complex Differential Equations. de Gruyter. ISBN 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 Several Complex Variables. American Mathematical Society. ISBN 978-0-8218-2961-5.

カテゴリ

[af-1] Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. en:Cambridge University Press.

[huku-2] 福原満洲雄「常微分方程式第2版」岩波全書.

[taka-3] 常微分方程式, 朝倉書店, 高野恭一.

[kim-4] 木村俊房「常微分方程式II」基礎数学講座.

[g2p-5] Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions. en:Springer Science & Business Media.

[6] 原岡喜重. (2002). 超幾何関数. 朝倉書店.

[7] 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.

[8] Bruno, A. D., & Batkhin, A. B. (Eds.). (2012). Painlevé Equations and Related Topics. de Gruyter.

[9] Noumi, M. (2004). Painlevé equations through symmetry. en:Springer Science & Business Media.

[10] Bobenko, A. I., Berlin, A. I. B. T., & Eitner, U. (2000). Painlevé equations in the differential geometry of surfaces. en:Springer Science & Business Media.

[sophia-11] 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.

[oka-12] 岡本和夫. (2009). パンルヴェ方程式. 岩波書店.

[13] 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.

[14] Maier, R. (2007). The 192 solutions of the Heun equation. en:Mathematics of Computation, 76(258), 811-843.

[15] Maier, R. S. (2005). On reducing the Heun equation to the hypergeometric equation. Journal of Differential Equations, 213(1), 171-203.

[16] Bittanti, S., Laub, A. J., & Willems, J. C. (Eds.). (2012). The Riccati Equation. Springer Science & Business Media.

[17] 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.

[18] Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory (2001), Zhidkov, Peter E., Springer.

[19] The Nonlinear Schrödinger Equation (1999) -Self-Focusing and Wave Collapse- , Sulem, Catherine, Sulem, Pierre-Louis, Springer.

[20] The Nonlinear Schrödinger Equation -Singular Solutions and Optical Collapse- (2015), Gadi Fibich, Springer.

[21] 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.

[22] 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.

[23] 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.

[24] Yuan, J. M., & Wu, J. (2005). The complex KdV equation with or without dissipation. Discrete Contin. Dyn. Syst. Ser. B, 5, 489-512.

[25] 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.

[26] Qi-Lao, Z., & Zhi-Bin, L. (2008). Darboux transformation and multi-solitons for complex mKdV equation. Chinese Physics Letters, 25(1), 8.

[27] 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.

[28] 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.

[29] Kenyon, R., & Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Mathematica, 199(2), 263-302.

[30] Liu, T. P., & Zumbrun, K. (1995). Nonlinear stability of an undercompressive shock for complex Burgers equation. Communications in mathematical physics, 168(1), 163-186.

[31] 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.

[32] Neuberger, H. (2008). Complex Burgers' equation in 2D SU (N) YM. Physics Letters B, 670(3), 235-240.

[33] 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.

[34] 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.

[35] Park, Q. H., & Shin, H. J. (1995). Duality in complex sine-Gordon theory. Physics Letters B, 359(1-2), 125-132.

[36] 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.

[37] 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.

[38] Park, Q. H., & Shin, H. J. (1999). Complex sine-Gordon equation in coherent optical pulse propagation. arXiv preprint solv-int/9904007.

[39] 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.

[40] 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.

[41] Aranson, I. S., & Kramer, L. (2002). The world of the complex Ginzburg-Landau equation. Reviews of Modern Physics, 74(1), 99.

[42] Akhmediev, N. N., Ankiewicz, A., & Soto-Crespo, J. M. (1997). Multisoliton solutions of the complex Ginzburg-Landau equation. Physical review letters, 79(21), 4047.

[43] Shraiman, B. I., Pumir, A., van Saarloos, W., 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.

[44] Van Saarloos, W., & Hohenberg, P. C. (1990). Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Physical review letters, 64(7), 749.

[45] Chate, H. (1994). Spatiotemporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation. Nonlinearity, 7(1), 185.

[46] 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.

[47] Hakim, V., & Rappel, W. J. (1992). Dynamics of the globally coupled complex Ginzburg-Landau equation. Physical Review A, 46(12), R7347.

[48] 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.

[49] Battogtokh, D., & Mikhailov, A. (1996). Controlling turbulence in the complex Ginzburg-Landau equation. Physica D: Nonlinear Phenomena, 90(1-2), 84-95.

[50] 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.

[51] M. Hukuhara, T. Kimura, T. Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe, Publ. Math. Soc. Japan, 7. The Mathematical Society of Japan, Tokyo 1961.

[52] 福原満洲雄. (1982). 常微分方程式の 50 年, II. 数学, 34(3), 262-269.

[53] 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).

[54] 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.

[55] 大島利雄. 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.

[56] 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).

[57] 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).

[58] 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.

[59] 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).

[60] 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).

[61] 岡本和夫. (1980). Painlevé の方程式. 数学, 32(1), 30-43.

[62] Takano, K., Reduction for Painlevé equations at the fixed singular points of the first kind, Funkcial. Ekvac., 29(1986), 99-119.

[63] Takano, K., Reduction for Painlevé equations at the fixed singular points of the second kind, J. Math. Soc. Japan, 42(1990), 423-443.

[64] 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.

[65] Shioda, T. and Takano, K., On some Hamiltonian structures of Painlevé systems, I, Funkcial. Ekvac., 40(1997), 271-291.

[66] Matano, T., Matumiya, A. and Takano, K., On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, 51(1999), 843-866.

[67] 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.

[68] Takano, K., Confluences of defining manifolds of Painlevé systems, Tohoku Math. J., 53(2001), 319-335.

[69] Noumi, M., Takano, K. and Yamada, Y., Bäcklund transformations and the manifolds of Painlevé systems, Funkcial. Ekvac., 45(2002), 237-258.

[70] Suzuki, M., Tahara, N. and Takano, K., Hierarchy of Bäcklund transformation groups of the Painlevé systems, J. Math. Soc. Japan, 56(2004), 1221-1232.

[71] Kimura, H. and Takano, K., On confluences of general hypergeometric systems, Tohoku Math. J., 58(2006), 1-31.

[72] Jimbo, M. (1982). Monodromy problem and the boundary condition for some Painlevé equations. Publications of the Research Institute for Mathematical Sciences, 18(3), 1137-1161.

[73] 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.

[74] Jimbo, M., Miwa, T., & Ueno, K. (1981). Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and $\tau$ -function. Physica D: Nonlinear Phenomena, 2(2), 306-352.

[75] 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.

[76] 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.

[its-77] 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. American Mathematical Society.

[78] Its, A. R., & Novokshenov, V. Y. (2006). The isomonodromic deformation method in the theory of Painlevé equations. Springer.

[79] Fokas, A. S., & Ablowitz, M. J. (1981). Linearization of the Korteweg—de Vries and Painlevé II Equations. Physical Review Letters, 47(16), 1096.

[80] 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.

[81] 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.

[82] 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.

[83] Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, 58, American Mathematical Society, ISBN 0821804960

[84] Trogdon, T., & Olver, S. (2015). Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions. SIAM.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

表話編歴数学
主要分野	数理論理学集合論圏論代数学初等線型多重線型抽象算術/数論解析学/微分積分学関数解析学行列解析幾何学代数微分有限離散位相代数的位相表現論リー理論（英語版）微分方程式線型微分方程式複素微分方程式常微分方程式偏微分方程式積分微分方程式力学系組合せ数学数理物理学ゲーム理論グラフ理論最適化問題数理最適化計算理論確率論数理統計学（英語版）制御理論三角法
トピックス	数学史和算算道数理哲学美未解決問題算数・数学教育算数教科
賞	ボッチャー記念賞コール賞フィールズ賞ヴェブレン賞サレム賞スティール賞ウルフ賞クラフォード賞クレイ研究賞ガウス賞アーベル賞ラマヌジャン賞 SASTRAラマヌジャン賞チャーン賞数学ブレイクスルー賞
応用	情報理論折紙の数学囲碁と数学数値解析精度保証付き数値計算計算機援用証明数値線形代数常微分方程式の数値解法偏微分方程式の数値解法
学会・団体	国際数学者会議国際数学連合国際産業数理・応用数理会議国際産業数理・応用数理評議会（英語版）アメリカ数学会 SIAM ヨーロッパ数学会ロンドン数学会日本応用数理学会日本数学会数学教育協議会
競技	国際数学オリンピック日本数学オリンピック日本ジュニア数学オリンピック広中杯近畿大学数学コンテスト人の最大の力を競う算数・数学の大会
研究所	京都大学数理解析研究所明治大学先端数理科学インスティテュート統計数理研究所アンリ・ポアンカレ研究所オーバーヴォルファッハ数学研究所クレイ研究所 CIMAT数学研究センターステクロフ数学研究所
一覧数学定数数関数図形数学記号数学者エポニムカテゴリポータル

複素微分方程式

主な複素微分方程式

主な複素常微分方程式

主な複素偏微分方程式

研究者

日本

海外

関連項目

出典

参考文献

Suggest as cover photo

Thank you for helping!

Install Wikiwand

Don't forget to rate us

Tell your friends about Wikiwand!

Enjoying Wikiwand?

Tell your friends and spread the love:

Your preferred languages

All languages

Follow Us

Don't forget to rate us

Our magic isn't perfect

Thank you for helping!

Oh no, there's been an error