查看“︁卡东穆塞夫-彼得韦亚斯维利方程”︁的源代码

[[File:Ile de ré.JPG|thumb|right|300px|这张在法国[[大西洋]]岸[[雷岛]]（RHE）鲸鱼灯塔拍摄的照片，显示浅海上田字形的椭圆余弦波列。这种浅水中的[[孤波]]可以由卡东穆夫-彼得韦亚斯维利方程模拟。]]
'''卡东穆塞夫-彼得韦亚斯维利方程'''(Kadomtsev-Petviashvili equation)，简称KP方程，是1970年苏联物理学家波里斯·卡东穆塞夫  和弗拉基米尔-彼得韦亚斯维利创立以模拟非线性波动的非线性偏微分方程<ref>Kodomtsev,B.B and Petviashivili V.I. On the stability of solitary waves in weakly dispersive media  Dokl. Akad Nauk SSSR 192 753-6(1970) Soviet Phys. Dok 15,539-41(1970)</ref>：

:<math>\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0</math>

其中 <math>\lambda=\pm 1</math>.

==解析解==
卡东穆塞夫-彼得韦亚斯维利方程有解析解<ref>Erk Infeld & George Rowlands, Nonlinear Waves,Solitons and Chaos p224-233 Cambridge University Press,2000</ref>

===行波解===
<math>u(x,y,t)= C5+12.*_C2*\tanh(_C1+_C2*x+_C3*y-(.50000000000000000000*(8.*_C2^4+_C3^2))*t/_C2)</math>

代人参数： C5 = 1, _C1 = 0, _C2 = 1, _C3 = 3

得：<math>u= 1+12.*tanh(x+3*y-8.5000000000000000000*t)</math>

{{Gallery
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|File:Kadomtsev-Petviahivili equation 3D plot.png|Kadomtsev-Petviahivili equation 3D plot
|File:Kadomtsev-Petviashivili pde animation.gif|Kadomtsev-Petviashivili pde animation
}}



===Sech 函数亮孤立子解===
利用sech函数展开法可得卡东穆塞夫-彼得韦亚斯维利方程的sech函数解和tanh函数解<ref>AHMET BEKIR and ÖZKAN GÜNER Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation,journal of Physics, August 2013 Vol. 81, No. 2, pp. 203–214</ref>。



<math>u := a*sech(a*x+b*y+c*z-(a^4+3*b^2+3*c^2)/a)*t</math>

参数：a = -2 .. 2, b = -2 .. 2, c = 0
{{Gallery
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|File:Kadomtsev Petviashivili pde sech solution 3d plot.png|Kadomtsev Petviashivili pde sech solution 3d plot
|File:Kadomtsev-Petviashivili pde bright soliton.gif|卡东穆塞夫-彼得韦亚斯维利方程sech函数亮孤立子
}}

===tanh 函数解===
<math>u := 2*a^2*tanh(a*x+b*y+(8*a^4-3*b^2)/a)^2*t</math><ref>AHMET BEKIR and ÖZKAN GÜNER </ref>。

参数：a = 2, b = -2；
{{Gallery
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|align=center
|File:Kadomtsev-Petviashivili pde dark soliton.gif|KP方程暗tanh函数暗孤立子
}}


===雅可比橢圓函數解===
通过[[朗斯基行列式]]展开法可得卡东塞穆夫-彼得韦亚斯维利方程多个雅可比橢圓函數解<ref>吕大昭等 Novel Interaction Solutions to Kadomtsev–Petviashvili Equation，Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 484–488</ref>。

<math>u4 := \frac{(-4*m^2*k[1]^2*g)}{(\sqrt{1-m^2}*sn(\xi[1], k)*sin(\xi[2])+dn(\xi[1], k)*cos(\xi[2])*cn(\xi[1], k))^2)}</math>

其中：

<math> g = (m^2-1)*sn(\xi[1], k)^2+(2-2*m^2)*sn(\xi[1], k)^4+cos(\xi[2])^2; -2*sn(\xi[1], k)^2*cos(\xi[2])^2+m^2*sn(\xi[1], k)^4*cos(\xi[2])^2</math>

<math> \xi[1] = k[1]*x+\lambda[1]*y+(4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1]</math>


<math>\xi[2] = \sqrt{1-m^2}*(k[1]*x+\lambda[1]*y+(-4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t)-\gamma[2]</math>

代入后得：

<math>f4 := -4*m^2*k[1]^2*((m^2-1)*JacobiSN(k[1]*x+\lambda[1]*y+(4*m^2*k[1]^3+16*k[1]^3</math>

<math>-3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1], k)^2+(2-2*m^2)*JacobiSN(k[1]*x+\lambda[1]*y</math>

<math>+(4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1], k)^4+</math>

<math>cos(\sqrt{(1-m^2)}*(k[1]*x+\lambda[1]*y+(-4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t)</math>

<math>-\gamma[2])^2)/(sqrt(1-m^2)*JacobiSN(k[1]*x+\lambda[1]*y+(4*m^2*k[1]^3+16*k[1]^3-</math>

<math>3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1], k)*sin(\sqrt(1-m^2)*(k[1]*x+\lambda[1]*y+</math>

<math>(-4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t)-\gamma[2])+JacobiDN(k[1]*x+\lambda[1]*y+</math>

<math>(4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1], k)*cos(\sqrt{1-m^2}*(k[1]*x+\lambda[1]*y+</math>

<math>(-4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t)-\gamma[2])*JacobiCN(k[1]*x+</math>

<math>\lambda[1]*y+(4*m^2*k[1]^3+16*k[1]^3-3*\sigma^2*\lambda[1]^2/k[1])*t+\gamma[1], k))^2</math>

{{Gallery
|width=270
|height=252
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|File:Kadomtsev Petviashivili pde elliptic function solution 3d plot.png|Kadomtsev Petviashivili pde elliptic function solution 3d plot
}}

==参考文献==
<references/>
# *谷超豪 《[[孤立子]]理论中的[[达布变换]]及其几何应用》 上海科学技术出版社
# *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
# 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
#王东明著 《消去法及其应用》 科学出版社 2002
# *何青 王丽芬编著 《[[Maple]] 教程》 科学出版社 2010 ISBN 9787030177445
#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
# Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
#Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004
# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759
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[[Category:非线性偏微分方程]]
[[category:孤立子]]