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)是一個非線性偏微分方程^[1]。

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

行波解

多德-布洛-米哈伊洛夫方程不是函數u的多項式形式，因此必須做代換：

$v=e^{u}$ , 變為：

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

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

作反變換： $u(x,t)=ln(v(x,t))$

即得多德-布洛-米哈伊洛夫方程的行波解：

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

行波圖

參考文獻

^ 李志斌編著《非線性數學物理方程的行波解》第105-107頁，科學出版社 2008

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李志斌編著《非線性數學物理方程的行波解》科學出版社
王東明著《消去法及其應用》科學出版社 2002
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[1] 李志斌編著《非線性數學物理方程的行波解》第105-107頁，科學出版社 2008

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