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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 ?

EditPiAf
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Michael
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    Looks pretty ugly. If $\alpha=\gamma=0$ and $\beta>0$, then symmetry helps a bit: a shock wave will form at $x=1$, and it will not go anywhere (by symmetry), with all characteristics eventually terminating in it. But in general with all these constants... even the qualitative picture will depend on the sign and magnitude of those. Generally, one find the moment of shock formation, and then follows the shock wave as it propagates with the speed $(v_++v_-)/2$ (average of speed to the left and right). But here this approach leads to an ODE that doesn't look easy to solve. –  Oct 27 '15 at 03:27
  • This post concerns the particular case $\alpha = 0 = \gamma$, $\beta = 1$. – EditPiAf Dec 05 '17 at 13:15

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