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Ablowitz, M. , Bakirtas, I. , and Ilan, B. , “Wave collapse in a class of nonlocal nonlinear Schrödinger equations,” Physica D 207, 230253 (2005).
Adams, R. , Sobolev Spaces (Academic, New York, 1975).
Cao, Y. , Musslimani, Z. H. , and Titi, E. S. , “Modulation theory for self-focusing in the nonlinear Schrödinger-Helmholtz equation,” Numer. Funct. Anal. Optim. 30, 4669 (2009).
Cao, Y. , Musslimani, Z. H. , and Titi, E. S. , “Nonlinear Schrödinger-Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation,” Nonlinearity 21, 879898 (2008).
Carles, R. , Semi-Classical Analysis for Nonlinear Schrödinger Equations (World Scientific Publishing Co. Pte. Ltd., Singapore, 2008).
Cazenave, T. , An Introduction to Nonlinear Schrödinger Equations (Instituto de Matemática-UFRJ, RJ, 1996).
Cazenave, T. , Semilinear Schrödinger Equations, Courant Lecture notes in Mathematics (American Mathematical Society, Providence, RI, 2003).
Cipolatti, R. , “On the existence of standing waves for the Davey-Stewartson system,” Commun. Partial Differ. Equations 17, 967988 (1992).
Davey, A. and Stewartson, K. , “On three dimensional packets of surface waves,” Proc. R. Soc. A 338, 101110 (1974).
Djordjevic, V. D. and Redekopp, L. G. , “On two-dimensional packets of capillary- gravity waves,” J. Fluid Mech. 79, 703714 (1977).
Eden, A. and Kuz, E. , “Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering,” Commun. Pure Appl. Anal. 8, 18031823 (2009).
Fibich, G. and Papanicolaou, G. , “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183240 (1999).
Ghidaglia, J. M. and Saut, J. C. , “On the initial value problem for the Davey-Stewartson systems,” Nonlinearity 3, 475506 (1990).
Ginibre, J. and Velo, G. , “On a class of nonlinear Schrödinger equation. I. The Cauchy problem, general case,” J. Funct. Anal. 32, 132 (1979).
Glassey, R. T. , “On the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equation,” J. Math. Phys. 18, 17941797 (1977).
Kato, T. , “On nonlinear Schrödinger equations,” Ann. Inst. H. Poincaré Phys. Théor. 46, 113129 (1987).
Kelley, P. L. , “Self-focusing of the optical beams,” Phys. Rev. Lett. 15, 10051008 (1965).
Mitrović, D. and Zubrinić, D. , Fundamentals of Applied Functional Analysis, Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 91 (Longmans Green, London, 1977).
Papanicolaou, G. C. , Sulem, C. , Sulem, P. L. , and Wang, X. P. , “The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface wave,” Physica D 72, 6186 (1994).
Stein, E. M. , Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, NJ, 1970).
Stein, E. M. and Shakarchi, R. , Functional Analysis: Introduction to Further Topics in Analysis (Princeton University Press, Princeton, NJ, 2011).
Sulem, C. and Sulem, P. L. , The Nonlinear Schrödinger Equation Self-Focsing and Wave Collapse, Applied Mathematical Sciences Vol. 139 (Springer-Verlag, Berlin, 1999).
Weinstein, M. I. , “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567576 (1983).
Yudovich, V. I. , “Non-stationary flow of an ideal incompressible liquid,” Zh. Vychils. Mat. 3, 10321066 (1963);
[Yudovich, V. I. , Comput. Math. Phys. 3, 14071456 (1963) (in English)].
Yudovich, V. I. , The Linearization Method in Hydrodynamical Stability Theory, Translations of Mathematical Monographs Vol. 74 (American Mathematical Society, Providence, RI, 1989).

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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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