KdV-Burgers方程

KdV-Burgers也称Burgers-KdV方程是一个非线性偏微分方程：^[1][2]

${\displaystyle u_{t}+u*u_{x}-\alpha *u_{xx}-\beta *u_{xxx}=0}$

解析解

${\displaystyle u(x,t)=(1/25)*(-3+250*\beta ^{2}*_{C}3)/\beta +(6/25)*coth(_{C}1-(1/10)*x/\beta +_{C}3*t)/\beta +(3/25)*coth(_{C}1-(1/10)*x/\beta +_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(1/25)*(-3+250*\beta ^{2}*_{C}3)/\beta +(6/25)*tanh(_{C}1-(1/10)*x/\beta +_{C}3*t)/\beta +(3/25)*tanh(_{C}1-(1/10)*x/\beta +_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=-(1/25)*(3+250*\beta ^{2}*_{C}3)/\beta -(6/25)*tanh(_{C}1+(1/10)*x/\beta +_{C}3*t)/\beta +(3/25)*tanh(_{C}1+(1/10)*x/\beta +_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(1/25)*(-(250*I)*\beta ^{2}*_{C}3-3)/\beta -(6/25*I)*tan(_{C}1-(1/10*I)*x/\beta +_{C}3*t)/\beta -(3/25)*tan(_{C}1-(1/10*I)*x/\beta +_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(1/25)*(-(250*I)*\beta ^{2}*_{C}3-3)/\beta +(6/25*I)*cot(_{C}1-(1/10*I)*x/\beta +_{C}3*t)/\beta -(3/25)*cot(_{C}1-(1/10*I)*x/\beta +_{C}3*t)^{2}/\beta }$

行波图

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot 2

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot

KdV-Burgers equation traveling wave plot

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参考文献

1. Shu, Jian-Jun. The proper analytical solution of the Korteweg-de Vries-Burgers equation. Journal of Physics A-Mathematical and General. 1987, 20 (2): 49–56. doi:10.1088/0305-4470/20/2/002.
2. 李志斌编著 《非线性数学物理方程的行波解》 61页 科学出版社 2008

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