查看“︁卡马萨-霍尔姆方程”︁的源代码


'''卡马萨-霍尔姆方程'''(Camassa Holm equation)是流体力学中的一个非线性偏微分方程

:<math>
u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. \,
</math>

1993年卡马萨和霍尔姆以此偏微分方程模拟浅水波<ref name=CH1993>Camassa & Holm 1993</ref>，

其中κ是大于0的参数。

==行波解==
[[File:Camassa Holm nlpde 3d plot.gif|thumb|right|350px|卡马萨-霍尔姆方程3D动画]]
卡马萨-霍尔姆方程有行波解<ref>Beals, Sattinger & Szmigielski 1999</ref>：

<math>u2 := (3/2)*\frac{c*(c-2*\kappa)}{(cosh((1/2)*\sqrt{((c-2*\kappa)/c)}*(-x-x0+c*t))^2*(\kappa+c))}</math>

参数：c = 1, x0 = 1, kappa = .3  代人得：

<math>u(x,t)=\frac{.463}{cosh(-.316*x-.316+.316*t)^2}</math>
===Maple TWSolution===
[[Maple]]软件包TWSolution可提供多种行波解<ref>Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27-35</ref>。
;sech 展开
<math>g[2] := {u(x, t) = -(1/2)*kappa+_C5*sech(_C1+(1/2)*sqrt(3)*x-(1/4)*kappa*sqrt(3)*t)^2}</math>
<math>g[3] := {u(x, t) = -(1/2)*kappa+_C5*sech(_C1-(1/2)*sqrt(3)*x+(1/4)*kappa*sqrt(3)*t)^2}</math>
<math>g[4] := {u(x, t) = _C4+(-(3/2)*_C4-(3/4)*kappa)*sech(_C1+(1/2*I)*sqrt(3)*x-(1/4*I)*kappa*sqrt(3)*t)^2}</math>
<math>g[5] := {u(x, t) = _C4+(-(3/2)*_C4-(3/4)*kappa)*sech(_C1-(1/2*I)*sqrt(3)*x+(1/4*I)*kappa*sqrt(3)*t)^2}</math>
<math>g[6] := {u(x, t) = -(_C3-4*_C3*_C2^2+2*kappa*_C2)/(_C2*(3+4*_C2^2))+24*_C2*(2*_C3+kappa*_C2)*sech(_C1+_C2*x+_C3*t)^2/(16*_C2^4-9)}</math>
{{Gallery
|width=250
|height=200
|align=center
|File:Camassa Holm equation traveling wave sech plot5.gif|Camassa Holm equation traveling wave sech plot5
|File:Camassa Holm equation traveling wave sech plot4.gif|Camassa Holm equation traveling wave sech plot4
|File:Camassa Holm equation traveling wave sech plot6.gif|Camassa Holm equation traveling wave sech plot6
}}
;exp  展开
<math>g[2] := {u(x, t) = -(1/9)*sqrt(3)*_C3-(1/3)*kappa+_C5*exp(_C1-sqrt(3)*x+_C3*t)}</math>
<math>g[3] := {u(x, t) = (1/9)*sqrt(3)*_C3-(1/3)*kappa+_C5*exp(_C1+sqrt(3)*x+_C3*t)}</math>
<math>g[5] := {u(x, t) = -(1/3)*sqrt(3)*_C3-(1/3)*kappa+_C7*(exp(_C1-(1/3)*sqrt(3)*x+_C3*t))^3}</math>
{{Gallery
|width=250
|height=200
|align=center
|File:Camassa Holm equation traveling wave exp plot2.gif|Camassa Holm equation traveling wave exp plot2
|File:Camassa Holm equation traveling wave exp plot3.gif|Camassa Holm equation traveling wave exp plot3
|File:Camassa Holm equation traveling wave exp plot5.gif|Camassa Holm equation traveling wave exp plot5
}}
;csch 展开
<math>g[2] := {u(x, t) = -(1/2)*kappa+_C5*csch(_C1+(1/2)*sqrt(3)*x-(1/4)*kappa*sqrt(3)*t)^2}</math>
<math>g[4] := {u(x, t) = _C4+((3/2)*_C4+(3/4)*kappa)*csch(_C1+(1/2*I)*sqrt(3)*x-(1/4*I)*kappa*sqrt(3)*t)^2}</math>
<math>g[6] := {u(x, t) = -(_C3-4*_C3*_C2^2+2*kappa*_C2)/(_C2*(4*_C2^2+3))-24*_C2*(2*_C3+kappa*_C2)*csch(_C1+_C2*x+_C3*t)^2/(16*_C2^4-9)}</math>
{{Gallery
|width=250
|height=200
|align=center
|File:Camassa Holm equation traveling wave csch plot2.gif|Camassa Holm equation traveling wave csch plot2
|File:Camassa Holm equation traveling wave csch plot4.gif|Camassa Holm equation traveling wave csch plot4
|File:Camassa Holm equation traveling wave csch plot6.gif|Camassa Holm equation traveling wave csch plot6
}}
;sec 展开
<math>g[3] := {u(x, t) = -(1/2)*kappa+_C5*sec(_C1-(1/2*I)*sqrt(3)*x+(1/4*I)*kappa*sqrt(3)*t)^2}</math>
<math>g[5] := {u(x, t) = _C4+(-(3/2)*_C4-(3/4)*kappa)*sec(_C1-(1/2)*sqrt(3)*x+(1/4)*kappa*sqrt(3)*t)^2}</math>

<math>g[6] := {u(x, t) = (_C3+4*_C3*_C2^2+2*kappa*_C2)/(_C2*(4*_C2^2-3))-24*_C2*(2*_C3+kappa*_C2)*sec(_C1+_C2*x+_C3*t)^2/(16*_C2^4-9)}</math>
{{Gallery
|width=250
|height=200
|align=center
|File:Camassa Holm equation traveling wave sec plot3.gif|Camassa Holm equation traveling wave sec plot3
|File:Camassa Holm equation traveling wave sec plot5.gif|Camassa Holm equation traveling wave sec plot5
|
}}
;JacobiSN 展开
<math>g[3] := {u(x, t) = (1/9*I)*sqrt(3)*_C4-(1/3)*kappa+_C6*sin(_C2-I*sqrt(3)*x+_C4*t)}</math>
<math>g[4] := {u(x, t) = -(2/9*I)*sqrt(3)*_C4-(1/2)*_C7-(1/3)*kappa+_C7*sin(_C2+(1/2*I)*sqrt(3)*x+_C4*t)^2}</math>

[[File:Camassa Holm equation traveling wave Jacobish plot4.gif|thumb|center|250px|Camassa Holm equation traveling wave Jacobish plot4]]
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>

==参考文献==
{{非线性偏微分方程}}
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# *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
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#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

[[category:非线性偏微分方程]]
[[category:孤立子]]