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。

u_{xt}+\alphae^u+\gammae^{-2*u} = 0

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：

v=e^u, 变为：

vv_{xt}-v_{t}v_{x}+\alpha*v^3+\gamma = 0

得到函数v(x,t)的行波解：
v(x, t) = (1/2)\gamma^(1/3)+(3/2)\gamma^(1/3)cot(_C1+_C2x-(3/4)\gamma^(1/3)t/_C2)^2
v(x, t) = (1/2)\gamma^(1/3)-(3/2)\gamma^(1/3)coth(_C1+_C2x+(3/4)\gamma^(1/3)t/_C2)^2
v(x, t) = (1/2)\gamma^(1/3)+(3/2)\gamma^(1/3)tan(_C1+_C2x-(3/4)\gamma^(1/3)t/_C2)^2
v(x, t) = (1/2)\gamma^(1/3)-(3/2)\gamma^(1/3)tanh(_C1+_C2x+(3/4)\gamma^(1/3)t/_C2)^2
v(x, t) = -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+(-(3/4)\gamma^(1/3)-(3/4I)\sqrt(3)\gamma^(1/3))cot(_C1+_C2x+(3/4)((1/2)\gamma^(1/3)+(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2
v(x, t) = -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+(-(3/4)\gamma^(1/3)-(3/4I)\sqrt(3)\gamma^(1/3))tan(_C1+_C2x+(3/4)((1/2)\gamma^(1/3)+(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2
v(x, t) = -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+((3/4)\gamma^(1/3)+(3/4I)\sqrt(3)\gamma^(1/3))coth(_C1+_C2x+(3/4)(-(1/2)\gamma^(1/3)-(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2
v(x, t) = -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+((3/4)\gamma^(1/3)+(3/4I)\sqrt(3)\gamma^(1/3))tanh(_C1+_C2x+(3/4)(-(1/2)\gamma^(1/3)-(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2

作反变换：
u(x,t)=ln(v(x,t))

即得多德-布洛-米哈伊洛夫方程的行波解：

u(x, t) =ln( (1/2)\gamma^(1/3)+(3/2)\gamma^(1/3)cot(_C1+_C2x-(3/4)\gamma^(1/3)t/_C2)^2)
u(x, t) =ln( (1/2)\gamma^(1/3)-(3/2)\gamma^(1/3)coth(_C1+_C2x+(3/4)\gamma^(1/3)t/_C2)^)
u(x, t) =ln( (1/2)\gamma^(1/3)+(3/2)\gamma^(1/3)tan(_C1+_C2x-(3/4)\gamma^(1/3)t/_C2)^2)
u(x, t) =ln( (1/2)\gamma^(1/3)-(3/2)\gamma^(1/3)tanh(_C1+_C2x+(3/4)\gamma^(1/3)t/_C2)^2)
u(x, t) =ln( -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+(-(3/4)\gamma^(1/3)-(3/4I)\sqrt(3)\gamma(1/3))cot(_C1+_C2x+(3/4)((1/2)\gamma^(1/3)+(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2)
u(x, t) =ln( -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+(-(3/4)\gamma^(1/3)-(3/4I)\sqrt(3)\gamma^(1/3))tan(_C1+_C2x+(3/4)((1/2)\gamma^(1/3)+(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2)
u(x, t) =ln( -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+((3/4)\gamma^(1/3)+(3/4I)\sqrt(3)\gamma^(1/3))coth(_C1+_C2x+(3/4)(-(1/2)\gamma^(1/3)-(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2)
u(x, t) =ln( -(1/4)\gamma^(1/3)-(1/4I)\sqrt(3)\gamma^(1/3)+((3/4)\gamma^(1/3)+(3/4I)\sqrt(3)\gamma^(1/3))tanh(_C1+_C2x+(3/4)(-(1/2)\gamma^(1/3)-(1/2I)\sqrt(3)\gamma^(1/3))t/_C2)^2)

行波图
参考文献

*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