複素微分方程式
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主な複素常微分方程式
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- 1 2 3 Ablowitz, M. J., & Fokas, A. S. (2003). Complex variables: introduction and applications. en:Cambridge University Press.
- 1 2 福原満洲雄「常微分方程式 第2版」岩波全書.
- 1 2 3 4 5 常微分方程式, 朝倉書店, 高野恭一.
- 1 2 木村俊房「常微分方程式II」基礎数学講座.
- 1 2 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.
- 1 2 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.
- 1 2 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
-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.
- 1 2 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
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- 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.
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