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/content/aip/journal/jmp/57/8/10.1063/1.4960047
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/content/aip/journal/jmp/57/8/10.1063/1.4960047
2016-08-02
2016-09-28

Abstract

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 -models of turbulence and therefore are called the -regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the -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 , Nonlinearity , 879–898 (2008); Cao , Numer. Funct. Anal. Optim. , 46–69 (2009)] on the non-viscous -regularization of the nonlinear Schrödinger equation.

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