查看“︁KdV方程”︁的源代码

'''科特韦赫-德弗里斯方程'''（{{lang-en|Korteweg-De Vries equation}}），一般简称'''KdV方程'''，是1895年由荷兰数学家[[迪德里克·科特韦赫|科特韦赫]]和[[古斯塔夫·德弗里斯|德弗里斯]]共同发现的一种[[偏微分方程]]。关于实自变量''x'' 和''t'' 的[[函数]]φ所满足的KdV方程形式如下：
:<math>\partial_t\phi-6\phi\partial_x\phi+\partial^3_x\phi=0</math>

KdV方程的解为簇集的[[孤立子]]（又称'''孤子'''，'''孤波'''）。

==KdV方程的行波解==
KdV 方程有多种孤波解<ref>阎振亚著 《复杂非线性波动构造性理论及其应用》 29页 科学出版社 2007</ref><ref>Graham W.Griffiths  William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p422-430</ref>。
*钟形孤波解

:<math>\phi(x,t)=\frac12\, c\, \mathrm{sech}^2\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]</math>
*扭形孤波解
:<math>\phi(x,t)=k\, \mathrm{tanh}[k(x+2tk^2+c)]</math><br />

*暗孤波解

<math>\phi(x,t)= a+b\, \mathrm{tanh}(1+cx+dt)^2</math>
<gallery>
File:Bellsoliton2.gif|钟形孤波解
File:Twister soliton.gif|扭形孤波解
File:Dark solition.gif|暗孤波解
</gallery>

==tanh 法解==
利用Maple tanh 法可得 孤立子解：<ref>Graham W.Griffiths William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p391-404</ref>。

:<math>{u(x, t) = (1/6)*(4*_C2^3-_C3)/_C2-2*_C2^2*csc(_C1+_C2*x+_C3*t)^2}  </math>
:<math> {u(x, t) = (1/6)*(4*_C2^3-_C3)/_C2-2*_C2^2*sec(_C1+_C2*x+_C3*t)^2} </math>
:<math> u(x, t) = -(1/6)*(4*_C2^3+_C3)/_C2-2*_C2^2*csch(_C1+_C2*x+_C3*t)^2 </math>
:<math> {u(x, t) = -(1/6)*(4*_C2^3+_C3)/_C2+2*_C2^2*sech(_C1+_C2*x+_C3*t)^2} </math>
:<math> {u(x, t) = (1/6)*(8*_C2^3-_C3)/_C2-2*_C2^2*coth(_C1+_C2*x+_C3*t)^2} </math>
:<math> {u(x, t) = (1/6)*(8*_C2^3-_C3)/_C2-2*_C2^2*tanh(_C1+_C2*x+_C3*t)^2} </math>
:<math> {u(x, t) = -(1/6)*(8*_C2^3+_C3)/_C2-2*_C2^2*cot(_C1+_C2*x+_C3*t)^2} </math>
:<math> {u(x, t) = -(1/6)*(8*_C2^3+_C3)/_C2-2*_C2^2*tan(_C1+_C2*x+_C3*t)^2} </math>
:<math>{u(x, t) = (1/6)*(-8*_C3^3+4*_C3^3*_C1^2-_C4)/_C3+2*_C3^2*JacobiDN(_C2+_C3*x+_C4*t, _C1)^2}  </math>
:<math> {u(x, t) = (1/6)*(-8*_C3^3+4*_C3^3*_C1^2-_C4)/_C3+(2*_C3^2-2*_C3^2*_C1^2)*JacobiND(_C2+_C3*x+_C4*t, _C1)^2} </math>
:<math> {u(x, t) = (1/6)*(4*_C3^3*_C1^2+4*_C3^3-_C4)/_C3-2*_C3^2*JacobiNS(_C2+_C3*x+_C4*t, _C1)^2} </math>
:<math> {u(x, t) = (1/6)*(4*_C3^3*_C1^2+4*_C3^3-_C4)/_C3-2*_C3^2*_C1^2*JacobiSN(_C2+_C3*x+_C4*t, _C1)^2} </math>
:<math> {u(x, t) = -(1/6)*(8*_C3^3*_C1^2-4*_C3^3+_C4)/_C3+(-2*_C3^2+2*_C3^2*_C1^2)*JacobiNC(_C2+_C3*x+_C4*t, _C1)^2} </math>
:<math> {u(x, t) = -(1/6)*(8*_C3^3*_C1^2-4*_C3^3+_C4)/_C3+2*_C3^2*_C1^2*JacobiCN(_C2+_C3*x+_C4*t, _C1)^2} </math>
:<math>9.81207-7.70406*I+5.44331*arctanh(10.4881/\sqrt(-110.*csc(1.40000+1.50000*x+1.60000*t)^2+110.))  </math>
:<math> 9.81207-7.70406*I-5.44331*arctan(10.4881/\sqrt(-110.*csch(1.40000+1.50000*x+1.60000*t)^2-110.)) </math>
:<math> 9.81207-7.70406*I+5.44331*arctan(10.4881/\sqrt(-110.*csch(1.40000+1.50000*x+1.60000*t)^2-110.)) </math>

==三维行波图==
{|
|[[File:KdV equation traveling wave plot 1.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 2.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 3.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 4.gif|thumb|KdV方程行波图]]
|}
{|
|[[File:KdV equation traveling wave plot 5.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 6.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 7.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 8.gif|thumb|KdV方程行波图]]
|}
{|
|[[File:KdV equation traveling wave plot 9.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 10.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 11.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 12.gif|thumb|KdV方程行波图]]
|}
{|
|[[File:KdV equation traveling wave plot 13.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 14.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 15.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 16.gif|thumb|KdV方程行波图]]
|}
{|
|[[File:KdV equation traveling wave plot 17.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 18.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 19.gif|thumb|KdV方程行波图]]
|[[File:KdV equation traveling wave plot 20.gif|thumb|KdV方程行波图]]
|}

== 联系 ==
KdV方程在物理学的许多领域都有应用，例如等离子体磁流波、离子声波、非谐振晶格振动、低温非线性晶格声子波包的热激发、液体气体混合物的压力表等。

KdV方程也可以用[[逆散射]]技术求解。

== 相关 ==
* [[格罗斯–皮塔耶夫斯基方程]]
* [[孤立子]]

== 延伸阅读 ==
* Korteweg, D. J. and de Vries, F. "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves." Philosophical Magazine, '''39''', 422--443, 1895.
* P. G. Drazin. ''Solitons''. Cambridge University Press, 1983.

== 参考文献 ==
{{Reflist}}
# *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

{{非线性偏微分方程}}

[[category:非线性偏微分方程]]
[[Category:精确解模型]]
[[Category:孤立子]]
[[Category:流体力学中的方程]]