Nonlinearity management in higher dimensions

In the present paper, we revisit nonlinearity management of the time-periodic nonlinear Schrodinger equation and the related averaging procedure. By means of rigorous estimates, we show that the averaged nonlinear Schrodinger equation does not blow up in the higher dimensional case so long as the corresponding solution remains smooth. In particular, we show that the H-1 norm remains bounded, in contrast with the usual blow-up mechanism for the focusing Schrodinger equation. This conclusion agrees with earlier works in the case of strong nonlinearity management but contradicts those in the case of weak nonlinearity management. The apparent discrepancy is explained by the divergence of the averaging procedure in the limit of weak nonlinearity management.