 Solution

For $0<t<1$, translated copies of the preceding pulse have disjoint interiors of support. In the cell $k-1/2\le x<k+1/2$, write $r=x-k$. The solution isAt $t=0$ use the given initial function. Each fan starts at $k-1/2$, and each shock has position $x=k+t/2$. At $t=1$, the plateau disappears: its left fan catches its right shock at $k+1/2$, which is also the starting edge of the next fan. The shocks must henceforth connect two nonconstant fan branches rather than a fan to a zero interval.