The possibility of self-acceleration of the water-wave pulse with a permanent envelope in the form of the nonlinear Airy function during its long propagation in deep water is experimentally and theoretically analyzed. This wave packet has amazing properties — accelerates without any external force, and preserves shape in a dispersive medium. The inverted Airy envelope wave function can propagate at velocity that is faster than the group velocity. We experimentally study the behavior of Airy water-wave pulses in a super-tank and long scaled propagation, to investigate its main properties, nonlinear effects and stability. Theoretical modeling analysis is based on the nonlinear Schrodinger equation. We investigate the scope of applicability, feasibility and stability conditions of nonlinear Airy wave trains in the deep water conditions; defining regimes of self-acceleration of the main pulse, immutability shape of Airy envelope; assessing the impact of nonlinearity and dissipation on the propagation of Airy waves. We analyzed the influence of the initial pulse characteristics on self-acceleration of wave packet and the stability of the envelope form. The anticipated results allow extending the physical understanding of the evolution of nonlinear dispersive waves in a wide range of initial conditions and at different spatial and temporal scales, from both theoretical and experimental points of view. Steep waves start to become an unstable, we observe spectrum widening and downshifting. Wave propagation is accompained by the intensive wave breaking and the generation of water-wave solitons.