Blow-up dynamics and spectral property in the L2-critical nonlinear Schrödinger equation in high dimensions

Abstract

We study stable blow-up dynamics in the L2-critical nonlinear Schrödinger (NLS) equation in high dimensions. First, we show that in dimensions d = 4 to d = 12 generic blow-up behavior confirms the log–log regime in our numerical simulations under the radially symmetric assumption, and asymptotic analysis, including the log–log rate and the convergence of the blow-up profiles to the rescaled ground state; this matches the description of the stable blow-up regime in the 2D cubic NLS equation.Next, we address the question of rigorous justification of the log–log dynamics in higher dimensions (), at least for the initial data with the mass slightly larger than the mass of the ground state, for which the spectral conjecture has yet to be proved, see Merle and Raphaël (2005 Ann. Math. 161 157–222) and Fibich et al (2006 Physica D 220 1–13). We give a numerically-assisted proof of the spectral property for the dimensions from d = 5 to d = 12, and a modification of it in dimensions . This, combined with previous results of Merle–Raphaël, proves the log–log stable blow-up dynamics in dimensions and radially stable for .