the distribution function.

It is also of interest to consider $f_n^{(1)}$ for $n\geq 2$. From Eq. (11) we obtain for $|n|\geq 2$ $$\frac{\partial f_n^{(1)}}{\partial t}+(in\omega +n^2\xi
)f_n^{(1)}=inh_n\Delta (t) \ ,$$ which does not have any contribution from the order parameter, $R$. For $\tau >t>0$, Eq. (20) yields $$f_n^{(1)}(\omega ,t)=inh_nd_0\exp [-(in\omega +n^2\xi )t]\ ,$$ for $0 < t < \tau $, which, similar to the first term on the right hand side of Eq. (16), also oscillates increasingly more rapidly with $\omega $ as $t$ increases. At time $t=\tau ^-$ Eq. (21) yields $$f_n^{(1)}(\omega ,\tau ^-)=inh_nd_0\exp [-(in\omega +n^2\xi )\tau
] \ ,$$ for $|n|\geq 2$.

C. Order $\epsilon ^2$
----------------------

Now proceeding to $\mathcal{O}(\epsilon ^2)$ and again (as done in obtaining Eq. (15)) taking the $e^{i\theta }$ component of Eq. (11), we have $$\frac{\partial f_1^{(2)}}{\partial t}+(i\omega +\xi
)f_1^{(2)}-\frac{1}{2}KR^{(2)*}=-i\left\{
\frac{K}{2i}f_2^{(1)}R^{(1)}-\Delta (t)\sum
^{+\infty}_{n=-\infty}h_{-(n-1)}f^{(1)}_n\right\}$$ where $R^{(1,2)*}(t)=\int ^{+\infty}_{-\infty}g(\omega
)f_1^{(1,2)}(\omega ,t)d\omega $. The above equation is linear in $f_1^{(2)}$ and is driven by several inhomogeneous terms appearing on the right hand side of Eq. (23) that are quadratic in first order quantities. Since we are interested in the components of $f_1^{(2)}$ that result in echoes, and since, by our previous discussion, we expect that the echoes 