KP方程式のソースを表示

'''KP方程式''' ({{lang-en-short|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。
:<math>\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0,\quad\lambda=\pm 1.</math>
[[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<ref>Wazwaz, A. M. (2008). Solitons and singular solitons for the Gardner–KP equation. Applied Mathematics and Computation, 204(1), 162-169.</ref><ref>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.</ref><ref>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.</ref><ref>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.</ref>
* KP-Boussinesq 方程式<ref>Wazwaz, A. M., & El-Tantawy, S. A. (2017). Solving the <math>(3+1)</math>-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dynamics, 88(4), 3017-3021.</ref><ref>Sun, B., & Wazwaz, A. M. (2018). General high–order breathers and rogue waves in the <math>(3+1)</math>-dimensional KP–Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 64, 1-13.</ref>
* Lax-KP 方程式<ref>Wazwaz, A. M. (2008). Multiple-soliton solutions for the Lax–Kadomtsev–Petviashvili (Lax–KP) equation. Applied Mathematics and computation, 201(1-2), 168-174.</ref>
* 超離散KP方程式 ({{lang-en-short|Ultradiscrete KP equation}})<ref>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.</ref><ref name="udkp">Shinzawa, N., & Hirota, R. (2003). The Bäcklund transformation equations for the ultradiscrete KP equation. Journal of Physics A: Mathematical and General, 36(16), 4667.</ref><ref name="bkp">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).</ref>

==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)]]<ref>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.</ref>
* [[:en:Mark J. Ablowitz]]<ref>Fokas, A. S., & Ablowitz, M. J. (1983). Method of solution for a class of multidimensional nonlinear evolution equations. Physical Review Letters, 51(1), 7.</ref><ref>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.</ref><ref>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.</ref>
===日本===
* [[薩摩順吉]]<ref name="hos"/><ref>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).</ref><ref>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.</ref><ref>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.</ref><ref name="#1">Matsukidaira, J., Satsuma, J., & Strampp, W. (1990). Conserved quantities and symmetries of KP hierarchy. Journal of mathematical physics, 31(6), 1426-1434.</ref><ref name="#1"/><ref>Kajiwara, K., Matsukidaira, J., & Satsuma, J. (1990). Conserved quantities of two-component KP hierarchy. Physics Letters A, 146(3), 115-118.</ref>
* [[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)<ref>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.</ref><ref>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.</ref><ref>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.</ref><ref>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.</ref><ref>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.</ref>
* [[広田良吾]]<ref name="udkp"/><ref name="bkp"/><ref name="hos">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.</ref><ref>{{Cite report|和書 |author=広田良吾 |year=2013 |url=https://doi.org/10.15017/27167 |title=KP差分方程式系とその解の構造 |volume=24AO-S3 |issue=7 |pages=49-57 |doi=10.15017/27167 |hdl=2324/27167 |publisher=九州大学応用力学研究所 |quote=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」}}</ref>
* [[和達三樹]]<ref>{{Cite journal |author=Ohkuma, Kenji; Wadati, Miki |year=1983 |title=The Kadomtsev-Petviashvili Equation: the Trace Method and the Soliton Resonances |journal= Journal of the Physical Society of Japan |ISSN=00319015 |publisher=日本物理学会 |volume=52 |issue=3 |pages=749-760 |doi=10.1143/jpsj.52.749 |CRID=1390001204185121408 |quote=MRID:702929 |BIBCODE=1983JPSJ...52..749O}}</ref>

==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==
{{reflist}}

==参考文献==
{{参照方法|date=2019年10月}}
* {{Cite journal|first1=B. B.|last1=Kadomtsev|first2=V. I.|last2=Petviashvili|title=On the stability of solitary waves in weakly dispersive media|journal=Sov. Phys. Dokl.|volume=15|year=1970|pages=539–541|bibcode = 1970SPhD...15..539K }}. Translation of {{cite journal| title=Об устойчивости уединенных волн в слабо диспергирующих средах | journal=Doklady Akademii Nauk SSSR | volume=192 | issue= | pages=753–756}}
* {{SpringerEOM|title=KP-equation|last=Previato|first=Emma|urlname=KP-equation}}
* 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===
* {{Cite journal|和書|author=塩田隆比呂 |date=1989 |url=https://doi.org/10.11429/sugaku1947.41.16 |title=KP方程式とSchottky問題 |journal=数学 |ISSN=0039470X |publisher=日本数学会 |volume=41 |issue=1 |pages=16-33 |doi=10.11429/sugaku1947.41.16 |CRID=1390282680042479744 |ref=harv}}
* {{Cite journal|和書|author=藤井信太郎, 京藤敏達, 西村仁嗣 |year=1996 |url=https://doi.org/10.2208/proce1989.43.31 |title=KP方程式を用いたマッハ反射の数値解析 |journal=海岸工学論文集 |volume=43 |pages=31-35 |doi=10.2208/proce1989.43.31 |CRID=1390282679526394496 |ISSN=0916-7897 |publisher=土木学会 |ref=harv}}
* {{PDFlink|[https://hakotama.jp/laboratory/works/public/2005-RIAM-16ME-S1-192.pdf 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」]}} Reports of RIAM Symposium No.16ME-S1, Physics and Mathematical Structures of Nonlinear Waves, Proceedings of a symposium held at Chikushi Campus, [[九州大学|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.
* {{Cite journal|和書|author=塩崎峻介, 村上洋一 |url=https://hdl.handle.net/2433/140667 |title=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) |journal=数理解析研究所講究録 |volume=1645 |pages=157-167 |hdl=2433/140667 |CRID=1050282677155323392 |ISSN=1880-2818 |publisher=京都大学数理解析研究所 |ref=harv}}
* {{Cite journal|和書 |author=及川正行 |date=2013-06 |url=https://www.nagare.or.jp/download/noauth.html?dd=assets/files/download/noauth/nagare/32-3/32-3rensai.pdf | format=PDF |title=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－ |journal=[https://www.nagare.or.jp/publication/nagare/archive/2013/3.html ながれ] |volume=32 |issue=3 |pages=251-266 |ISSN=02863154 |publisher=日本流体力学会 |ref=harv}}

===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==
* {{mathworld|urlname=Kadomtsev-PetviashviliEquation|title=Kadomtsev–Petviashvili equation}}
* {{scholarpedia | urlname=Kadomtsev-Petviashvili_equation | title=Kadomtsev–Petviashvili equation | curator=Gioni Biondini and Dmitri Pelinovsky}}
* {{URL|http://www.amath.washington.edu/~bernard/kp.html}} (The KP page by Bernard Deconinck, [[ワシントン大学 (ワシントン州)|University of Washington]], Department of Applied Mathematics)
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[[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]]