$m$ of the order $O(10^{-6})$.

The core of the numerical scheme is the integration of the partial differential equations (\[SP2\]) and (\[SP3\]) along the $x$ direction which, at difference with previous numerical studies [@topomoto; @B90], is done in the Fourier Space of the $y$ variables where the equations take the form: $$\label{eq:mk}
 \frac{\partial}{\partial x}\,m(x,k) =
          \frac{k^2\dot{q}(x)}{2}\, m(x,k)
         -\frac{\beta \dot{q}(x)}{2}\, \mbox{i}k\, {\cal FT}\left[
              m^2\right](x,k)$$ and $$\label{eq:mP}
 \frac{\partial}{\partial x}\,P(x,k) =
         -\frac{k^2\dot{q}(x)}{2}\, P(x,k)
         -\beta \dot{q}(x)\, \mbox{i}k\, {\cal FT}\left[
              P\,m\right](x,k)$$ For each wave-vector $k$ these are ordinary differential equations which can be integrated using standard methods. To avoid the time consuming calculation of the convolutions in the non-linear term we use a pseudo-spectral[@Orsz] code on a grid mesh of $N_x\,\times\, N_y$ points, which covers the $x$-interval $[0,x_0]$ and the $y$-interval $[-y_{\rm max},y_{\rm max}]$. The truncation of wave-number may introduce anisotropic effects for large $k$, therefore to ensure a better isotropy of numerical treatment we perform de-aliasing via a $N_y/2$ truncation [@deal]. Finally the $x$ integration has been performed using an third-order Adam-Bashfort scheme which has the advantage of reducing the number of Fast Fourier calls [@AB]. Typical values used are $N_x=100 \div 5000$, $N_y = 512 \div 4096$ and $y_{\rm max} = 12 \div 