Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation

Abstract

A regularized α-system of the nonlinear Schrödinger (NLS) equation with 2σ nonlinear power in dimension N is studied. We prove short time existence and uniqueness of solution in the case 1 σ <. And in the case 1 σ < 3 (when N = 1) or in the case 1 σ < 4 N (when N > 1) we show global in time existence of solutions. When α → 0 + , the solutions of this regularized system will converge to the solutions of the classical NLS in the appropriate range when the latter exists. Consequently, we propose this regularized system as a numerical regularization to shed light on the profile of the blow-up solutions of the original NLS equation in the range 2 N σ < 4 N , and in particular for the classical critical case N = 2, σ = 1. Following the modulation theory, we derive the reduced system of ordinary differential equations for the Schrödinger-Helmholtz (SH) system. We observe that the reduced equations for this SH system are more complicated than the equations of some other perturbation regularizations of the classical NLS equation. The detailed analysis of the reduced system on how the regularization prevents singularity formation will be presented in a forthcoming paper.

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

“…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 [3,4] …”

“…Hence, in this section, we will apply modulation theory to the RDS3 system (3.5) by following the ideas in [12,19,22] (see also [3]) for the purpose of observing the prevention mechanism of the singularities. First, we review some main results on an asymptotic construction of blow-up solutions for the DSE presented in [19,22].…”

“…These regularizations are in the spirit of the α−models of turbulence. We will follow the approach in [3,4] in which an α-regularized nonlinear Schrödinger equation (NLS) was investigated. See also references to the α-models of turbulence in [3].…”

“…On the other hand, the DSE can be regarded as a perturbation of the 2d cubic NLS, and this perturbation does not effect the blow up rate [12,19]. In [3,4], the following non-viscous regularized system of the cubic NLS equation (1.4) is investigated:…”

“…Notice that when α = 0, (1.5) reduces to (1.4). It is shown in [3] that the Cauchy problem (1.5) is globally well-posed. Moreover, by regarding system (1.5) as a perturbation of the cubic NLS equation (1.4), and by adopting the modulation theory, different scenarios are demonstrated in [4] of how the regularization prevents the formation of the singularities of the cubic NLS equation.…”