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

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

海外

日本

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

^ 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.
^ Wazwaz, A. M., & El-Tantawy, S. A. (2017). Solving the $(3+1)$ -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 $(3+1)$ -dimensional KP–Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 64, 1-13.
^ Wazwaz, A. M. (2008). Multiple-soliton solutions for the Lax–Kadomtsev–Petviashvili (Lax–KP) equation. Applied Mathematics and computation, 201(1-2), 168-174.
^ 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.
^ ^a ^b Shinzawa, N., & Hirota, R. (2003). The Bäcklund transformation equations for the ultradiscrete KP equation. Journal of Physics A: Mathematical and General, 36(16), 4667.
^ ^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).
^ 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.
^ Fokas, 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.
^ ^a ^b 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.
^ 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.
^ ^a ^b 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.
^ 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『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。https://doi.org/10.15017/27167。「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」」
^ 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. CRID 1390001204185121408. doi:10.1143/jpsj.52.749. ISSN 00319015. "MRID:702929"

参考文献

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。

外部リンク

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）。

[1] Wazwaz, A. M. (2008). Solitons and singular solitons for the Gardner–KP equation. Applied Mathematics and Computation, 204(1), 162-169.

[2] 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.

[3] 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.

[4] 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.

[5] Wazwaz, A. M., & El-Tantawy, S. A. (2017). Solving the $(3+1)$ -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dynamics, 88(4), 3017-3021.

[6] Sun, B., & Wazwaz, A. M. (2018). General high–order breathers and rogue waves in the $(3+1)$ -dimensional KP–Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 64, 1-13.

[7] Wazwaz, A. M. (2008). Multiple-soliton solutions for the Lax–Kadomtsev–Petviashvili (Lax–KP) equation. Applied Mathematics and computation, 201(1-2), 168-174.

[8] 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.

[udkp-9] Shinzawa, N., & Hirota, R. (2003). The Bäcklund transformation equations for the ultradiscrete KP equation. Journal of Physics A: Mathematical and General, 36(16), 4667.

[bkp-10] 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.

[hos-15] 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.

[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.

[#1-19] Matsukidaira, 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。https://doi.org/10.15017/27167。「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. CRID 1390001204185121408. doi:10.1143/jpsj.52.749. ISSN 00319015. "MRID:702929"

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KP方程式

変種

KP方程式に関連した業績のある研究者

海外

日本

関連項目

出典

参考文献

関連文献

和文

英文

外部リンク