Non-homogeneous Burgers' equation

Question

If I have a complex Burgers equation $$u_x=uu_y+C,$$ where $x,y$ are real numbers.

Can anyone explain me what is the method of 'complex characteristic'? Is it differ from usual method of characteristic? or the literature about it..

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{du}{dt}=C$ , letting $u(0)=u_0$ , we have $u=u_0+Ct=u_0+Cx$

$\dfrac{dy}{dt}=-u=-u_0-Ct$ , letting $y(0)=f(u_0)$ , we have $y=f(u_0)-u_0t-\dfrac{Ct^2}{2}=f(u-Cx)-(u-Cx)x-\dfrac{Cx^2}{2}=f(u-Cx)+\dfrac{Cx^2}{2}-ux$ , i.e. $u=Cx+F\left(\dfrac{Cx^2}{2}-ux-y\right)$

As the above procedure also works on complex $u$ , $x$ and $y$ , so 'complex characteristic' has no difference between usual 'real characteristic'.