# KP方程式

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

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。

## 出典

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 ${\displaystyle (3+1)}$ -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 ${\displaystyle (3+1)}$ -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.
9. ^ 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.
10. ^ 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 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.
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 b 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. (2013). KP 差分方程式系とその解の構造, 京都大学数理解析研究所講究録
27. ^ Ohkuma, K., & Wadati, M. (1983). The Kadomtsev-Petviashvili equation: the trace method and the soliton resonances. Journal of the Physical society of Japan, 52(3), 749-760.

## 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. Bibcode1970SPhD...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 (ed.), Encyclopaedia 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

### 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.