Non-viscous Regularization of the Davey-Stewartson Equations: Analysis and Modulation Theory

TL;DR

This paper introduces non-viscous regularizations for the elliptic-elliptic Davey-Stewartson equations, proving their global well-posedness and analyzing how they prevent finite-time singularities using modulation theory.

Contribution

It proposes three novel non-viscous regularization systems for DSE, establishes their global well-posedness, and uses modulation theory to understand how these regularizations prevent singularities.

Findings

Regularized systems are globally well-posed for all initial data.

Regularizations prevent finite-time blow-up in the elliptic-elliptic DSE.

Modulation theory helps analyze the singularity prevention mechanism.

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 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.