projection factor, which intervenes in the r.h.s. of (\[projdyn\]) to keep $\vec b$ on a sphere : $$A(\tau)=\frac{<\vec{N}[\vec{b}](\tau),\vec{b}(\tau)>}
{<\vec{b}(\tau),\vec{b}(\tau)>}
\label{defA}$$ will be of central importance in the following.\
Characteristic time scales on shell $n$ are in first approximation proportional to $b_{n}$ but now $b_{n}$ cannot exceed the initial value of $\sqrt{<\vec{b},\vec{b}>}$. It follows that within the “projected dynamics” defined by Eq. (\[projdyn\]), the cascade towards small scales is not accompanied by an acceleration of motion as in the original equations. There is now no impediment against taking a very large number of shells since the required time resolution does not grow anymore exponentially with $N$. By integrating numerically (\[projdyn\]), we observed that any initial condition of finite support (i.e. $b_{n}(0)\neq 0$ for $0\leq n \leq
n_{0}$, with $n_{0}\ll N$) gives birth at large times $\tau$ to a solitary wave moving with a constant velocity towards small scales. In other words, a period $T$ may be defined such that asymptotically, for $\tau
\rightarrow +\infty$ (a more precise condition reads $1\ll \tau \ll NT$, because some reflection will ultimately occur on the ultraviolet boundary), $$b_{n+1}(\tau +T) =  b_{n}(\tau) \;\;\equiv \;\;
b_{n}(\tau)=b(n-\frac{\tau}{T})
\label{asymptb_n}$$ Note that (\[asymptb\_n\]) implies $A(\tau +T)= A(\tau)$. The shape of 