多德-布洛-米哈伊洛夫方程

**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}+\alpha*e^u+\gamma*e^{-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：

v*v_{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+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2
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
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
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
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
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
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
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

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+_C2*x-(3/4)*\gamma^(1/3)*t/_C2)^2)
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)^)
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)
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)
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)
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)
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)
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)




## 行波图

## 参考文献

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