  In the present study we are interested in the Davey-Stewartson equations
(DSE) that model packets of surface and capillary-gravity waves. We focus on
the elliptic-elliptic case, for which it is known that DSE may develop a
finite-time singularity. We propose three systems of non-viscous regularization
to the DSE in variety of parameter regimes under which the finite blow-up of
solutions to the DSE occurs. We establish the global well-posedness of the
regularized systems for all initial data. The regularized systems, which are
inspired by the $\alpha$-models of turbulence and therefore are called the
$\alpha$-regularized DSE, are also viewed as unbounded, singularly perturbed
DSE. Therefore, we also derive reduced systems of ordinary differential
equations for the $\alpha$-regularized DSE by using the modulation theory to
investigate the mechanism with which the proposed non-viscous regularization
prevents the formation of the singularities in the regularized DSE. This is a
follow-up of the work of Cao, Musslimani and Titi on the non-viscous
$\alpha$-regularization of the nonlinear Schr\"odinger equation.
