JOURNAL OF MATHEMATICAL PHYSICS, cilt.57, sa.8, 2016 (SCI İndekslerine Giren Dergi)
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 a variety of parameter
regimes under which the finite-time 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 α-models of turbulence and
therefore are called the α-regularized DSE, are also viewed as unbounded, singularly
perturbed DSE. Therefore, we also derive reduced systems of ordinary differential
equations for the α-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
[Cao et al., Nonlinearity 21, 879–898 (2008); Cao et al., Numer. Funct. Anal. Optim.
30, 46–69 (2009)] on the non-viscous α-regularization of the nonlinear Schrödinger
equation