KP方程式

KP方程式 (英: Kadomtsev–Petviashvili equation) 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。

\displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0,\quad \lambda =\pm 1.

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[編集]

Gardner-KP 方程式^[1]^[2]^[3]^[4]
KP-Boussinesq 方程式^[5]^[6]
Lax-KP 方程式^[7]
超離散KP方程式 (英: Ultradiscrete KP equation)^[8]^[9]^[10]

KP方程式に関連した業績のある研究者[編集]

海外[編集]

日本[編集]

薩摩順吉^[15]^[16]^[17]^[18]^[19]^[19]^[20]
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)^[21]^[22]^[23]^[24]^[25]
広田良吾^[9]^[10]^[15]^[26]
和達三樹^[27]

出典[編集]

(一)^ Wazwaz, A. M. (2008). Solitons and singular solitons for the Gardner–KP equation. Applied Mathematics and Computation, 204(1), 162-169. (二)^ 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. (三)^ 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. (四)^ 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. (五)^ Wazwaz, A. M., & El-Tantawy, S. A. (2017). Solving the

(3+1)

(3+1)

-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dynamics, 88(4), 3017-3021. (六)^ Sun, B., & Wazwaz, A. M. (2018). General high–order breathers and rogue waves in the

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-dimensional KP–Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 64, 1-13. (七)^ Wazwaz, A. M. (2008). Multiple-soliton solutions for the Lax–Kadomtsev–Petviashvili (Lax–KP) equation. Applied Mathematics and computation, 201(1-2), 168-174. (八)^ 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. (九)^ ^a ^bShinzawa, N., & Hirota, R. (2003). The Bäcklund transformation equations for the ultradiscrete KP equation. Journal of Physics A: Mathematical and General, 36(16), 4667. (十)^ ^a ^b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). (11)^ 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. (12)^ Fokas, A. S., & Ablowitz, M. J. (1983). Method of solution for a class of multidimensional nonlinear evolution equations. Physical Review Letters, 51(1), 7. (13)^ 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. (14)^ 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. (15)^ ^a ^bHirota, 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. (16)^ 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). (17)^ 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. (18)^ 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. (19)^ ^a ^bMatsukidaira, J., Satsuma, J., & Strampp, W. (1990). Conserved quantities and symmetries of KP hierarchy. Journal of mathematical physics, 31(6), 1426-1434. (20)^ Kajiwara, K., Matsukidaira, J., & Satsuma, J. (1990). Conserved quantities of two-component KP hierarchy. Physics Letters A, 146(3), 115-118. (21)^ 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. (22)^ 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. (23)^ 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. (24)^ 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. (25)^ 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. (26)^ 広田良吾﹃KP差分方程式系とその解の構造﹄︵レポート︶ 24AO-S3、7号、九州大学応用力学研究所、2013年、49-57頁。doi:10.15017/27167。hdl:2324/27167。"九州大学応用力学研究所研究集会報告 No.23AO-S7 ﹁非線形波動研究の進展 : 現象と数理の相互作用﹂"。 (27)^ Ohkuma, Kenji; Wadati, Miki (1983). “The Kadomtsev-Petviashvili Equation: the Trace Method and the Soliton Resonances”. Journal of the Physical Society of Japan (日本物理学会) 52 (3): 749-760. doi:10.1143/jpsj.52.749. ISSN 00319015. "MRID:702929"

参考文献[編集]

●Kadomtsev, B. B.; Petviashvili, V. I. (1970). “On the stability of solitary waves in weakly dispersive media”. Sov. Phys. Dokl. 15: 539–541. Bibcode: 1970SPhD...15..539K.. Translation of “Об устойчивости уединенных волн в слабо диспергирующих средах”. Doklady Akademii Nauk SSSR 192: 753–756. ●Previato, Emma (2001), “KP-equation”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 ●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[編集]

和文[編集]

●塩田隆比呂﹁KP方程式とSchottky問題﹂﹃数学﹄第41巻第1号、日本数学会、1989年、16-33頁、CRID 1390282680042479744、doi:10.11429/sugaku1947.41.16、ISSN 0039470X。 ●藤井信太郎, 京藤敏達, 西村仁嗣﹁KP方程式を用いたマッハ反射の数値解析﹂﹃海岸工学論文集﹄第43巻、土木学会、1996年、31-35頁、CRID 1390282679526394496、doi:10.2208/proce1989.43.31、ISSN 0916-7897。 ●応用力学研究所研究集会報告 No.16ME-S1﹁非線形波動の物理と数理構造﹂ (PDF) 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. ●塩崎峻介, 村上洋一﹁KP方程式における線ソリトンの不安定性に関する直接数値計算 (非線形波動現象の数理と応用)﹂﹃数理解析研究所講究録﹄第1645巻、京都大学数理解析研究所、157-167頁、CRID 1050282677155323392、hdl:2433/140667、ISSN 1880-2818。 ●及川正行﹁第8章KP方程式のソリトン解 : 連載—非線形波動－ソリトンを中心として－﹂︵PDF︶﹃ながれ﹄第32巻第3号、日本流体力学会、2013年6月、251-266頁、ISSN 02863154。

英文[編集]

●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[編集]

●Weisstein, Eric W. "Kadomtsev–Petviashvili equation". mathworld.wolfram.com (英語). ●Gioni Biondini and Dmitri Pelinovsky (ed.). "Kadomtsev–Petviashvili equation". Scholarpedia. ●www.amath.washington.edu/~bernard/kp.html (The KP page by Bernard Deconinck, University of Washington, Department of Applied Mathematics)

この項目は、解析学に関連した書きかけの項目です。この項目を加筆・訂正などしてくださる協力者を求めています（プロジェクト:数学／Portal:数学）。

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