Melt spinning of the fibers by supersonic air jet in the Laval nozzle is a novel, efficient and energy saving method of formation of super-thin fibers. In the process, polymer melt is extruded from a row of orifices and fast drawn by the pneumatic forces. In the modelling, air velocity, temperature and pressure distributions are computed from the k-! aerodynamic model. Computations of the polymer air-drawing dynamics are based on the mathematical model of melt spinning in a single-, thin-filament approximation and Phan-Thien/Tanner non-linear viscoelasticity of the polymer melt. Axial profiles of the polymer velocity, temperature, tensile stress and rheological extra-pressure are computed. Influence of the Laval nozzle geometry, initial air compression, an initial melt temperature, a polymer mass output and the diameter of the melt extrusion die is discussed. The role of the polymer molecular weight, melt viscosity and relaxation time is considered. Example computations show the influence of important processing and material parameters. In the supersonic process, a high negative internal extra-pressure is predicted in the polymer melt under high elongation rates which may lead to cavitation and longitudinal burst splitting of the filament into a high number of sub-filaments. A hypothetical number of sub-filaments at the splitting is estimated from an energetic criterion. The diameter of the sub-filaments may reach the range of nano-fibers. A substantial influence of the Laval nozzle geometry is also predicted.