伯格斯方程

伯格斯方程（Burgers equation）是一个模拟冲击波的传播和反射的非线性偏微分方程，给定函数 ${\displaystyle u(x,t)}$ ，与扩散系数 ${\displaystyle \nu }$ ，则一维的伯格斯方程可以写成以下的形式:

${\displaystyle {\frac {\partial u(x,t)}{\partial t}}+u(x,t){\frac {\partial (u(x,t))}{\partial x}}=\nu {\frac {\partial ^{2}(u(x,t))}{\partial x^{2}}}}$

Backlund 变换

伯格斯方程通过下列Backlund变换^[1]。

${\displaystyle w[x]={\frac {1}{2}}wu}$

${\displaystyle w[t]={\frac {1}{2}}w[x]u+{\frac {1}{2}}wu[x]}$

则化为 热方程：

${\displaystyle HEq:=w[t]=w[x,x]}$

解析解

伯格斯方程有钟行孤波和扭波两种形式的行波解^[2]

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot

Burgers equation traveling wave plot