for $t<0$, and, in writing Eq. (12), $f_1$ represents the $e^{i\theta }$ component of the Fourier expansion of $f(\omega ,\theta ,t)$ in $\theta $, $$f(\omega ,\theta ,t)=\sum ^{+\infty}_{n=-\infty }f_n(\omega
,t)e^{in\theta } \ ,$$ with $f_0=1$, $f_n=f^*_{-n}$. As seen in Eq. (11), the effect of the noise term in Eq. (1) is to introduce diffusion in the phase angle $\theta $ whose strength is characterized by the phase diffusion coefficient $\xi $.

In Sec. III we will solve Eqs. (11) and (12) for the case $d_p\ll
1$, thus demonstrating the echo phenomenon as described in Sec. I. In Sec. IV we will present numerical solutions of Eq. (1) for large $N$.

III. Analysis
=============

A. Amplitude expansion
----------------------

In order to proceed analytically we use a small amplitude expansion and obtain results to second order (i.e., up to quadratic in the small amplitude). This will be sufficient to obtain the echo phenomenon. We introduce a formal expansion parameter $\epsilon $, as follows, $$f=1+\epsilon f^{(1)}+\epsilon ^2f^{(2)}+\mathcal{O}(\epsilon ^3) \
;$$ $\hat d_p=\epsilon d_p $ for $p=0$, $1$; $R=\epsilon
R^{(1)}+\epsilon ^2R^{(2)}+\mathcal{O}(\epsilon ^3)$; $R^{(m)*}=\int gf_1^{(m)}d\omega $; where $f^{(m)}=\Sigma
_nf_n^{(m)}\exp (in\theta )$. (Although we formally take $\epsilon
\ll 1$, when we finally get our answers, the 