查看“︁多德-布洛-米哈伊洛夫方程”︁的源代码

'''多德-布洛-米哈伊洛夫方程'''(Dodd-Bullough-Mikhailov equation)是一个非线性偏微分方程<ref>李志斌编著 《非线性数学物理方程的行波解》  第105-107页，科学出版社 2008</ref>。

<math>u_{xt}+\alpha*e^u+\gamma*e^{-2*u} = 0</math>

==行波解==
多德-布洛-米哈伊洛夫方程不是函数u的多项式形式，因此必须做代换：

<math>v=e^u</math>, 变为：

<math>v*v_{xt}-v_{t}*v_{x}+\alpha*v^3+\gamma = 0</math>

得到函数v(x,t)的行波解：
<math>v(x, t) = (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*cot(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2</math>
<math>v(x, t) = (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*coth(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2</math>
<math>v(x, t) = (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*tan(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2</math>
<math>v(x, t) = (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*tanh(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2</math>
<math>v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*cot(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2</math>
<math>v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*tan(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2</math>
<math>v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*coth(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2</math>
<math>v(x, t) = -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*tanh(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2</math>

作反变换：
<math>u(x,t)=ln(v(x,t))</math>

即得多德-布洛-米哈伊洛夫方程的行波解：

<math>u(x, t) =ln( (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*cot(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2)</math>
<math>u(x, t) =ln( (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*coth(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^)</math>
<math>u(x, t) =ln( (1/2)*\gamma^(1/3)+(3/2)*\gamma^(1/3)*tan(_C1+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2)</math>
<math>u(x, t) =ln( (1/2)*\gamma^(1/3)-(3/2)*\gamma^(1/3)*tanh(_C1+_C2*x+(3/4)*\gamma^(1/3)*t/_C2)^2)</math>
<math>u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma(1/3))*cot(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)</math>
<math>u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+(-(3/4)*\gamma^(1/3)-(3/4*I)*\sqrt(3)*\gamma^(1/3))*tan(_C1+_C2*x+(3/4)*((1/2)*\gamma^(1/3)+(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)</math>
<math>u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*coth(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)</math>
<math>u(x, t) =ln( -(1/4)*\gamma^(1/3)-(1/4*I)*\sqrt(3)*\gamma^(1/3)+((3/4)*\gamma^(1/3)+(3/4*I)*\sqrt(3)*\gamma^(1/3))*tanh(_C1+_C2*x+(3/4)*(-(1/2)*\gamma^(1/3)-(1/2*I)*\sqrt(3)*\gamma^(1/3))*t/_C2)^2)</math>


<math></math>
<math></math>

==行波图==
{|
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot10.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot14.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot15.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot16.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|}
{|
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot17.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot19.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot2.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot20.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|}
{|
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot21.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot3.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot4.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot5.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|}
{|
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot8.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|[[File:Dodd-Bullough-Mikhailov equation traveling wave plot9.gif|thumb|Dodd-Bullough-Mikhailov equation traveling wave plot]]
|
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|}

==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]]