Criteria for Collapse in the Focusing Nonlinear Schrodinger Equation

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 energy-critical NLS equations for some initial data u(0) with E[u(0)] > E[W], where W is the stationary solution to Delta W + W4/N-2 - 0. Furthermore, we prove that the initial data of the form W(x)e(i gamma) vertical bar(x)vertical bar(2) blows up if gamma < 0 and scatters if gamma > 0 in dimension 7 and higher. These collapse criteria are also applicable in the case of the energy-supercritical focusing NLS equation.