伯格斯方程

伯格斯方程（Burgers equation）是一個模擬衝擊波的傳播和反射的非線性偏微分方程，給定函數 $u(x,t)$ ，與擴散係數 $\nu$ ，則一維的伯格斯方程可以寫成以下的形式:

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

Backlund 變換

伯格斯方程通過下列Backlund變換^[1]。

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

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

則化為熱方程：

$HEq:=w[t]=w[x,x]$

伯格斯方程有鐘行孤波和扭波兩種形式的行波解^[2]

Burgers equation traveling wave plot

^ Inna Shingareva, Carlos Lizarraga Celaya, Solving Nonlinear Differential Equations with Maple and Mathematica, p47,Springer
^ Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p57-58Springer

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