Yanping Cao et al 2008 Nonlinearity 21 879 doi:10.1088/0951-7715/21/5/001
Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation
Yanping Cao1, Ziad H Musslimani2 and Edriss S Titi3,4
Show affiliationsRecommended by F Merle
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 
. And in the case 1 ≤ σ < 3 (when N = 1) or in the case 
(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 
, 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.
- PACS
- MSC
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35Q55 NLS-like (nonlinear Schrödinger) equations (See also 37K10)
37K40 Soliton theory, asymptotic behavior of solutions
35Q51 Solitons (See also 37K40)
35J05 Laplace equation, reduced wave equation (Helmholtz), Poisson equation (See also 31Axx, 31Bxx)
- Subjects
- Dates
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Issue 5 (May 2008)
Received 27 June 2007, in final form 7 February 2008
Published 26 March 2008
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Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation
Yanping Cao et al 2008 Nonlinearity 21 879
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