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Non-viscous regularization of the Davey-Stewartson equations: Analysis and modulation theory

### 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 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.

Published by AIP Publishing.

Received 06 May 2016
Accepted 14 July 2016
Published online 02 August 2016

Acknowledgments:
I.H. would like to thank the University of California–Irvine for the kind hospitality, where part of this work was completed. The work of I.H. was partially supported by TÜBİTAK (Turkish Scientific and Technological Research Council). The work of E.S.T. was supported in part by the ONR Grant No. N00014–15–1–2333 and the NSF Grant Nos. DMS–1109640, and DMS–1109645.

Article outline:

I. INTRODUCTION
II. NOTATIONS AND PRELIMINARIES
III. HELMHOLTZ *α*-REGULARIZED DAVEY-STEWARTSON EQUATIONS
A. Case 1: *ρ* > 0 and *β* > 0
B. Case 2: *ρ* < *β* < 0
C. Case 3: *ρ* < 0 and *β* > 0
IV. PROOF OF THE GLOBAL WELL-POSEDNESS OF THE *α*-REGULARIZED DAVEY-STEWARTSON EQUATIONS
A. Short-time existence and uniqueness of solutions in *H*^{1}
B. Continuous dependence on initial data in *H*^{1}
C. Short-time existence and uniqueness of solutions in *H*^{2}
D. Conservation of the hamiltonian
E. The extension to global solutions in *H*^{1}
V. MODULATION THEORY

/content/aip/journal/jmp/57/8/10.1063/1.4960047

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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 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.

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