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 x t + α ∗ e u + γ ∗ e − 2 ∗ u = 0 {\displaystyle u_{xt}+\alpha *e^{u}+\gamma *e^{-2*u}=0} 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 {\displaystyle v=e^{u}} , 变为： v ∗ v x t − v t ∗ v x + α ∗ v 3 + γ = 0 {\displaystyle 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 ) ∗ γ ( 1 / 3 ) + ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ c o t ( C 1 + C 2 ∗ x − ( 3 / 4 ) ∗ γ ( 1 / 3 ) ∗ t / C 2 ) 2 {\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}} v ( x , t ) = ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ c o t h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ γ ( 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}} v ( x , t ) = ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ t a n ( C 1 + C 2 ∗ x − ( 3 / 4 ) ∗ γ ( 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}} v ( x , t ) = ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ t a n h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ γ ( 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}} v ( x , t ) = − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( − ( 3 / 4 ) ∗ γ ( 1 / 3 ) − ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ c o t ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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}} v ( x , t ) = − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( − ( 3 / 4 ) ∗ γ ( 1 / 3 ) − ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ t a n ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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}} v ( x , t ) = − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( ( 3 / 4 ) ∗ γ ( 1 / 3 ) + ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ c o t h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( − ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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}} v ( x , t ) = − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( ( 3 / 4 ) ∗ γ ( 1 / 3 ) + ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ t a n h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( − ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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： u ( x , t ) = l n ( v ( x , t ) ) {\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： u ( x , t ) = l n ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ c o t ( C 1 + C 2 ∗ x − ( 3 / 4 ) ∗ γ ( 1 / 3 ) ∗ t / C 2 ) 2 ) {\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})} u ( x , t ) = l n ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ c o t h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ γ ( 1 / 3 ) ∗ t / C 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)^{)}} u ( x , t ) = l n ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ t a n ( C 1 + C 2 ∗ x − ( 3 / 4 ) ∗ γ ( 1 / 3 ) ∗ t / C 2 ) 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})} u ( x , t ) = l n ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 3 / 2 ) ∗ γ ( 1 / 3 ) ∗ t a n h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ γ ( 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})} u ( x , t ) = l n ( − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( − ( 3 / 4 ) ∗ γ ( 1 / 3 ) − ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ c o t ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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})} u ( x , t ) = l n ( − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( − ( 3 / 4 ) ∗ γ ( 1 / 3 ) − ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ t a n ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( ( 1 / 2 ) ∗ γ ( 1 / 3 ) + ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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})} u ( x , t ) = l n ( − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( ( 3 / 4 ) ∗ γ ( 1 / 3 ) + ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ c o t h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( − ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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})} u ( x , t ) = l n ( − ( 1 / 4 ) ∗ γ ( 1 / 3 ) − ( 1 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) + ( ( 3 / 4 ) ∗ γ ( 1 / 3 ) + ( 3 / 4 ∗ I ) ∗ ( 3 ) ∗ γ ( 1 / 3 ) ) ∗ t a n h ( C 1 + C 2 ∗ x + ( 3 / 4 ) ∗ ( − ( 1 / 2 ) ∗ γ ( 1 / 3 ) − ( 1 / 2 ∗ I ) ∗ ( 3 ) ∗ γ ( 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 } 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 *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. 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