Solve the Burgers' equation: $$u_t+(\frac{u^2}{2})_x=0,\quad 0<x<2,\quad 0<t<\infty,$$ with periodic boundary conditions and the initial condition $$u(x,0)=\alpha+\beta\sin(\pi x+\gamma),\quad 0<x<2,$$ where $\alpha,\beta,\gamma$ are constants.
I can solve this problem using method of characteristic, but only before the time at which the characteristic cross and a shock forms. It seems that the method of characteristic does not work for shock? And I also read some materials about shocks and know how to solve the Riemann problem. However, I do not know how this problem after the shock forms ?