Burgers' equation $u_t + uu_x =0$ with $u(x,0)=x$

Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,0)=x$.

I am not sure on how to find the solution $u(x,t)$. I have learned the method of characteristics. I am neither sure on how to use the side condition in Burgers' equation.

Solution 1:

This PDE is called the (inviscid) Burgers' equation. The characteristic equations are the system of ODEs $${dx\over dt}=u, \quad {du\over dt}=0,$$ which has solution $$u=C_1, \quad x=ut+C_2,$$ where $C_1,C_2$ are constants. But $C_1$ is a function of $C_2$, so $$u=C_1\implies u=C_1(C_2)=C_1(x-ut).$$ To determine $C_1$, note that the side condition requires $$u(x,0)=C_1(x-u(x,0)\cdot 0)=x\implies C_1(x)=x.$$ Thus, $u=x-ut$ and we conclude $$u(x,t)={x\over 1+t}.$$