  In this study we suggest new approach to turbulence modeling. To develop this
approach, we construct the set of the vortex-like dynamical systems evolving in
the space $E_3$. These systems are constructed using the AKNS hierarchy so that
they generalize thin closed vortex filament described in the local induction
approximation. We construct and investigate the Hamiltonian dynamics of these
systems in terms of the non - standard variables. We formulate the principles
of quantization these systems focusing on the quantum description of the
filaments with linearized dynamics. To describe the energy of vortex-like
structures, we apply a new method based on the use of the central extended
Galilei group. In a special case, we calculate the energy levels of a system of
quantized vortices. The proposed approach makes it possible to describe both
the interaction of such vortex - like filaments and the processes of their
creation and annihilation in a natural way. As a result, we have constructed a
quantum turbulent flow model that demonstrates the relationship between
stochastic and integrable dynamics.
