KP方程式

KP方程式 () は非線形波動・水面波を記述する偏微分方程式であり、次のように表わされる。

KdV方程式の2次元版方程式であり、KdV方程式と並ぶ可積分系・ソリトン方程式の代表例である。

変種
 Gardner-KP 方程式Wazwaz, A. M. (2008). Solitons and singular solitons for the Gardner–KP equation. Applied Mathematics and Computation, 204(1), 162-169.Xu, B., & Liu, X. Q. (2009). Classification, reduction, group invariant solutions and conservation laws of the Gardner-KP equation. Applied mathematics and computation, 215(3), 1244-1250.Naz, R., Ali, Z., & Naeem, I. (2013). Reductions and new exact solutions of ZK, Gardner KP, and modified KP equations via generalized double reduction theorem. In Abstract and Applied Analysis (Vol. 2013). Hindawi.Jawad, A. J. A. M., Mirzazadeh, M., & Biswas, A. (2015). Dynamics of shallow water waves with Gardner–Kadomtsev–Petviashvili equation. Discrete and Continuous Dynamical Systems, Series S, 8(6), 1155-1164.
 KP-Boussinesq 方程式Wazwaz, A. M., & El-Tantawy, S. A. (2017). Solving the -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dynamics, 88(4), 3017-3021.Sun, B., & Wazwaz, A. M. (2018). General high–order breathers and rogue waves in the -dimensional KP–Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 64, 1-13.
 Lax-KP 方程式Wazwaz, A. M. (2008). Multiple-soliton solutions for the Lax–Kadomtsev–Petviashvili (Lax–KP) equation. Applied Mathematics and computation, 201(1-2), 168-174.
 超離散KP方程式 ()Tokihiro, T., Takahashi, D., & Matsukidaira, J. (2000). Box and ball system as a realization of ultradiscrete nonautonomous KP equation. Journal of Physics A: Mathematical and General, 33(3), 607.Shinzawa, N., & Hirota, R. (2003). The Bäcklund transformation equations for the ultradiscrete KP equation. Journal of Physics A: Mathematical and General, 36(16), 4667.新沢信彦, & 広田良吾. (2003). 超離散 KP 方程式, 超離散 BKP 方程式の Backlund 変換方程式 (可積分系研究の新展開: 連続・離散・超離散).

KP方程式に関連した業績のある研究者
海外
 セルゲイ・ノヴィコフ (数学者)Krichever, I. M., & Novikov, S. P. (1978). Holomorphic bundles over Riemann surfaces and the Kadomtsev—Petviashvili equation. I. Functional Analysis and Its Applications, 12(4), 276-286.
 :en:Mark J. AblowitzFokas, A. S., & Ablowitz, M. J. (1983). Method of solution for a class of multidimensional nonlinear evolution equations. Physical Review Letters, 51(1), 7.Fokas, A. S., & Ablowitz, M. J. (1983). On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation. Physics Letters A, 94(2), 67-70.Fokas, A. S., & Ablowitz, M. J. (1983). On the Inverse Scattering of the Time‐Dependent Schrödinger Equation and the Associated Kadomtsev‐Petviashvili (I) Equation. Studies in Applied Mathematics, 69(3), 211-228.
日本
 薩摩順吉松木平淳太, & 薩摩順吉. (1989). KP hierarchy の対称性と保存量 (ソリトン理論における広田の方法).Willox, R., Tokihiro, T., & Satsuma, J. (1997). Darboux and binary Darboux transformations for the nonautonomous discrete KP equation. Journal of Mathematical Physics, 38(12), 6455-6469.Isojima, S., Willox, R., & Satsuma, J. (2002). On various solutions of the coupled KP equation. Journal of Physics A: Mathematical and General, 35(32), 6893.Matsukidaira, J., Satsuma, J., & Strampp, W. (1990). Conserved quantities and symmetries of KP hierarchy. Journal of mathematical physics, 31(6), 1426-1434.Kajiwara, K., Matsukidaira, J., & Satsuma, J. (1990). Conserved quantities of two-component KP hierarchy. Physics Letters A, 146(3), 115-118.
 佐藤幹夫 (数学者)-佐藤理論
 神保道夫 (三輪哲二、柏原正樹などとの共同研究)Date, E., Jimbo, M., Kashiwara, M., & Miwa, T. (1982). Transformation groups for soliton equations—Euclidean Lie algebras and reduction of the KP hierarchy—. Publications of the Research Institute for Mathematical Sciences, 18(3), 1077-1110.Date, E., Jimbo, M., Kashiwara, M., & Miwa, T. (1981). Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–. Journal of the Physical Society of Japan, 50(11), 3806-3812.Date, E., Jimbo, M., Kashiwara, M., & Miwa, T. (1982). Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type. Physica D: Nonlinear Phenomena, 4(3), 343-365.Date, E., Jimbo, M., Kashiwara, M., & Miwa, T. (1982). Quasi-Periodic Solutions of the Orthogonal KP Equation—Transformation Groups for Soliton Equations V—. Publications of the Research Institute for Mathematical Sciences, 18(3), 1111-1119.Date, E., Jimbo, M., Kashiwara, M., & Miwa, T. (1981). KP hierarchies of orthogonal and symplectic type–Transformation groups for soliton equations VI–. Journal of the Physical Society of Japan, 50(11), 3813-3818.
 広田良吾Hirota, R., Ohta, Y., & Satsuma, J. (1988). Solutions of the Kadomtsev-Petviashvili equation and the two-dimensional Toda equations. Journal of the Physical Society of Japan, 57(6), 1901-1904.
 和達三樹

関連項目
 流体力学
 可積分系
 ソリトン
 ラックス・ペア
 広田の方法
 KdV方程式

出典

参考文献

 . Translation of 

 Kodama, Y. (2017). KP Solitons and the Grassmannians: combinatorics and geometry of two-dimensional wave patterns. Springer.
 時弘哲治、箱玉系の数理、朝倉書店。

関連文献
和文


  Reports of RIAM Symposium No.16ME-S1, Physics and Mathematical Structures of Nonlinear Waves, Proceedings of a symposium held at Chikushi Campus, Kyushu University, Kasuga, Fukuoka, Japan, November 15 - 17, 2004. Article No. 32: KP方程式による孤立波相互作用とRogue Waveの関連について, 辻英一，及川正行, A. V. Porubov.



英文
 Lou, S. Y., & Hu, X. B. (1997). Infinitely many Lax pairs and symmetry constraints of the KP equation. Journal of Mathematical Physics, 38(12), 6401-6427.
 Nakamura, A. (1989). A bilinear N-soliton formula for the KP equation. Journal of the Physical Society of Japan, 58(2), 412-422.
 Kodama, Y. (2004). Young diagrams and N-soliton solutions of the KP equation. Journal of Physics A: Mathematical and General, 37(46), 11169.
 Xiao, T., & Zeng, Y. (2004). Generalized Darboux transformations for the KP equation with self-consistent sources. Journal of Physics A: Mathematical and General, 37(28), 7143.
 Minzoni, A. A., & Smyth, N. F. (1996). Evolution of lump solutions for the KP equation. Wave Motion, 24(3), 291-305.

外部リンク


  (The KP page by Bernard Deconinck, University of Washington, Department of Applied Mathematics)

category:物理学の方程式
category:偏微分方程式
category:可積分系
category:振動と波動
Category:数学に関する記事