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、）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Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. :en:Cambridge University Press.。
主な複素微分方程式
主な複素常微分方程式

 超幾何微分方程式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.
 パンルヴェ方程式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.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.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.
 ホイン函数#ホインの方程式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.
 リッカチの微分方程式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.

主な複素偏微分方程式

 非線形シュレディンガー方程式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.
 複素KdV方程式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.
 複素mKdV方程式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.
 複素バーガース方程式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.
 複素短パルス方程式 ()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.
 複素サイン・ゴルドン方程式 ()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.
 複素 Ginzburg-Landau 方程式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.

研究者
日本

 福原満洲雄福原満洲雄「常微分方程式 第2版」岩波全書.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.
 高野恭一常微分方程式, 朝倉書店, 高野恭一.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.
 木村俊房木村俊房「常微分方程式II」基礎数学講座.
 神保道夫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 -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.

海外

 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 ItsFokas, 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.
 :en:Mark J. AblowitzFokas, 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.

関連項目
 フックス型微分方程式
 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, 
 :en:Riemann–Hilbert problemTrogdon, T., & Olver, S. (2015). Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions. SIAM.
 :en:Riemann–Hilbert correspondence

出典

参考文献

 Einar Hille (1976). Ordinary Differential Equations in the Complex Domain. Wiley. ., 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. .
 Ilpo Laine (1992). Nevanlinna Theory and Complex Differential Equations. de Gruyter. .
 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. .

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