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^[1]。

${\displaystyle u_{xt}+\alpha *e^{u}+\gamma *e^{-2*u}=0}$

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：

${\displaystyle v=e^{u}}$, 变为：

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

得到函数v(x,t)的行波解： ${\displaystyle v(x,t)=(1/2)*\gamma ^{(}1/3)+(3/2)*\gamma ^{(}1/3)*cot(_{C}1+_{C}2*x-(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=(1/2)*\gamma ^{(}1/3)-(3/2)*\gamma ^{(}1/3)*coth(_{C}1+_{C}2*x+(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=(1/2)*\gamma ^{(}1/3)+(3/2)*\gamma ^{(}1/3)*tan(_{C}1+_{C}2*x-(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=(1/2)*\gamma ^{(}1/3)-(3/2)*\gamma ^{(}1/3)*tanh(_{C}1+_{C}2*x+(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+(-(3/4)*\gamma ^{(}1/3)-(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*cot(_{C}1+_{C}2*x+(3/4)*((1/2)*\gamma ^{(}1/3)+(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+(-(3/4)*\gamma ^{(}1/3)-(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*tan(_{C}1+_{C}2*x+(3/4)*((1/2)*\gamma ^{(}1/3)+(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+((3/4)*\gamma ^{(}1/3)+(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*coth(_{C}1+_{C}2*x+(3/4)*(-(1/2)*\gamma ^{(}1/3)-(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2}}$ ${\displaystyle v(x,t)=-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+((3/4)*\gamma ^{(}1/3)+(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*tanh(_{C}1+_{C}2*x+(3/4)*(-(1/2)*\gamma ^{(}1/3)-(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2}}$

作反变换： ${\displaystyle u(x,t)=ln(v(x,t))}$

即得多德-布洛-米哈伊洛夫方程的行波解：

${\displaystyle u(x,t)=ln((1/2)*\gamma ^{(}1/3)+(3/2)*\gamma ^{(}1/3)*cot(_{C}1+_{C}2*x-(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln((1/2)*\gamma ^{(}1/3)-(3/2)*\gamma ^{(}1/3)*coth(_{C}1+_{C}2*x+(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{)}}$ ${\displaystyle u(x,t)=ln((1/2)*\gamma ^{(}1/3)+(3/2)*\gamma ^{(}1/3)*tan(_{C}1+_{C}2*x-(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln((1/2)*\gamma ^{(}1/3)-(3/2)*\gamma ^{(}1/3)*tanh(_{C}1+_{C}2*x+(3/4)*\gamma ^{(}1/3)*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln(-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+(-(3/4)*\gamma ^{(}1/3)-(3/4*I)*{\sqrt {(}}3)*\gamma (1/3))*cot(_{C}1+_{C}2*x+(3/4)*((1/2)*\gamma ^{(}1/3)+(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln(-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+(-(3/4)*\gamma ^{(}1/3)-(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*tan(_{C}1+_{C}2*x+(3/4)*((1/2)*\gamma ^{(}1/3)+(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln(-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+((3/4)*\gamma ^{(}1/3)+(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*coth(_{C}1+_{C}2*x+(3/4)*(-(1/2)*\gamma ^{(}1/3)-(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2})}$ ${\displaystyle u(x,t)=ln(-(1/4)*\gamma ^{(}1/3)-(1/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3)+((3/4)*\gamma ^{(}1/3)+(3/4*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*tanh(_{C}1+_{C}2*x+(3/4)*(-(1/2)*\gamma ^{(}1/3)-(1/2*I)*{\sqrt {(}}3)*\gamma ^{(}1/3))*t/_{C}2)^{2})}$

${\displaystyle }$ ${\displaystyle }$

行波图

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot

Dodd-Bullough-Mikhailov equation traveling wave plot