查看“︁川原方程”︁的源代码

[[File:Kawahara pde Maple animation.gif|right|200px]]
[[File:Kawahara pde Maple plot.png|thumb|right|200px|川原方程图]]
[[File:Kawahara pde Maple 3d plot.png|thumb|right|200px|川原方程 Maple 3D 图]]
'''川原方程'''（Kawahara pde）是一个非线性偏微分方程：<ref> Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p197-238 Equations Academy Press</ref>

<math>\frac{\partial u}{\partial t}+u*\frac{\partial u}{\partial x}+a*\frac{\partial^3 u}{\partial x^3}=b*\frac{\partial^5 u}{\partial x^5}</math>

广义川原方程有如下形式：<ref> Inna Shingareva, Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica , Springer, p192</ref>

<math>\frac{\partial u}{\partial t}+(1+u^2)*\frac{\partial u}{\partial x}+a*\frac{\partial^3 u}{\partial x^3}=b*\frac{\partial^5 u}{\partial x^5}</math>

==行波解==
川原方程有孤立子解：<ref>李志斌  非线性数学物理方程的行波解  第137页  科学出版社</ref>

<math>u(x,t)=\frac{105*a^2}{169*b*cosh^4(z)}+C1</math>

<math>z=\frac{1}{2}*k*x-(18*b*k^5+C1*k)*t</math>

<math>k=\sqrt{{\frac{a}{13*b}}}</math>

==tanh 行波解==
利用tanh 法，可得<ref>Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations p391-409 Academy Press</ref>


<math>u=\frac{-c (-1+\tanh(1/2 (c (-x-x0+c t))}{(a)))}</math>

==Maple 软件包行波解==
利用[[Maple]]的TWSolutions软件包可得广义川原方程的一系列行波解：<ref>Inna Shingareva et al  Solving Nonlinear Partial Differential Equations with Maple and Mathematica  Springer, p 192</ref>  
[[File:Kawahara pde Maple TWSolution.JPG]]

==行波图==
{|
|[[File:Kawahara equation traveling wave plot 1.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 2.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 3.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 4.gif|thumb|Kawahara equation traveling wave plot]]
|}
{|
|[[File:Kawahara equation traveling wave plot 5.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 6.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 7.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 8.gif|thumb|Kawahara equation traveling wave plot]]
|}
{|
|[[File:Kawahara equation traveling wave plot 9.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 10.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 11.gif|thumb|Kawahara equation traveling wave plot]]
|[[File:Kawahara equation traveling wave plot 12.gif|thumb|Kawahara equation traveling wave 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

{{非线性偏微分方程}}
[[category:非线性偏微分方程]]