  We study the non linear gravitational clustering of collisionless particles
in an expanding background using an integro-differential equation for the
gravitational potential. In particular, we address the question of how the
nonlinear mode-mode coupling transfers power from one scale to another in the
Fourier space if the initial power spectrum is sharply peaked at a given scale.
We show that the dynamical equation allows self similar evolution for the
gravitational potential $\phi_{\bf k}(t)$ in Fourier space of the form
$\phi_{\bf k}(t) = F(t)D({\bf k}) $ where the function $F(t)$ satisfies a
second order non-linear differential equation. We provide a complete analysis
of the relevant solutions of this equation thereby determining the asymptotic
time evolution of the gravitational potential and density contrast. The
analysis shows that both $F(t) $ and $D({\bf k})$ have well defined asymptotic
forms indicating that the power transfer leads to a universal power spectrum at
late times. The analytic results are compared with numerical simulations
showing good agreement.
