$K_c=2$), $\hat d_0=\hat
d_1$, $K=1$, $\tau =50$, and $\xi =0$. At $t=0^{-}$ we initialize each phase $\theta _i$ for $i=1,2,\ldots ,N$ randomly and independently with a uniform distribution in the interval $(0,2\pi
)$. We then apply the mapping given by Eq. (7) with $\hat d_p=\hat
d_0$ to each $\theta _i$ in order to simulate the effect of the delta function at $t=0$. Next we integrate Eq. (1) for each $i=1,2,\ldots ,N$ forward in time to $t=\tau ^{-}$, again apply the mapping Eq. (7) (but now with $\hat d_p=\hat d_1$), and we then continue the integration. At each time step we also calculate $R(t)$ using Eq. (8). Figure 2 shows results for $\hat d_0=\hat
d_1=1/4$, and

![$|R(t)|$ versus $t$ for (a) $N=10^6$, (b) $N=10^5$, (c) $N=10^4$, and (d) $N=10^3$, showing the echo at $t\cong 2\tau $ and the increase of fluctuations at lower $N$.](fig2)

![$|R(t)|$ versus $t$ blown up around $t\cong 2\tau =200$, for $N=10^6$, $10^5$, $10^4$, $10^3$ (solid curves) showing the increase of fluctuations at lower $N$. The dotted curve is the theoretical result from Eq. (39) with $\xi =0$.](fig3)

$$h(\theta )=\sin \theta +\sin 2\theta \ ,$$ for several different system sizes, $N=10^6$, $10^5$, and $10^4$. Figure 2(a–c) shows $|R(t)|$ 