Criteria for Collapse in the Focusing Nonlinear Schrödinger Equation
Abstract
Using the classical mechanics approach and rigidity type arguments on the variance identity, we obtain several new sufficient criteria for collapse for any L^{2}supercritical focusing NLS equation with finite (positive) energy and finite variance initial data, which are different from the classical blow up arguments. In particular, we show that these criteria produce collapsing solutions to the energycritical NLS equations for some initial data u_{0} with E[u_{0}]>E[W], where W is the stationary solution to ∆W+W^{4/N2} = 0. Furthermore, we prove that the initial data of the form W(x)e^{iγx2} blows up if γ<0 and scatters if γ>0 in dimension 7 and higher. These collapse criteria are also applicable in the case of the energysupercritical focusing NLS equation.
 Publication:

Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics
 Pub Date:
 September 2011
 DOI:
 10.1063/1.3636831
 Bibcode:
 2011AIPC.1389..717D
 Keywords:

 momentum;
 interpolation;
 energy conservation;
 data analysis;
 45.20.df;
 02.60.Ed;
 45.20.dh;
 07.05.Kf;
 Momentum conservation;
 Interpolation;
 curve fitting;
 Energy conservation;
 Data analysis: algorithms and implementation;
 data management