Quasi-solitons and asymptotic multiscaling in shell models of turbulence

A variation principle is suggested to find self-similar solitary solutions (solitons'') of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(0.001). It is found how the soliton shape, propagation time $t_n$ and the dynamical exponent $z_0$ (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of $z$ the author discovered quasi-solitons’’ which approximate with high accuracy corresponding self-similar equations for an interval of times from $-\mathrm{\infty }$ to some time in the vicinity of the peak maximum or even after it. The conjecture is that the trajectories in the vicinity of the quasi-solitons (with continuous spectra of $z$) provide an essential contribution to the multiscaling statistics of high-order correlation functions, referred as an ``asymptotic multiscaling’'. This contribution may be even more important than that of the trajectories in the vicinity of the exact soliton with a fixed value $z_0$. Moreover there are no solitons in some region of the parameters where quasi-solitons provide dominant contribution to the asymptotic multiscaling