Logarithmic concentration for the focusing cubic Schrödinger equation on the two-dimensional torus

Abstract

We construct a smooth solution of the focusing cubic nonlinear Schrödinger equation on a square two-dimensional torus which is global in both time directions and concentrates at one point as time tends to positive infinity. Its mass is the mass of the Euclidean ground state and its energy is zero. If $λ(t)$ denotes its concentration scale and $L$ the period, then $L/λ(t)=2\log t+\frac{11}{2}\log\log t+O(1)$. The solution approaches a phased, periodized ground state with an error tending to zero in every fixed Sobolev space. In particular, its $H^s$ norm is asymptotic to an explicit constant times $(\log t)^s$ for every $s>0$. We also prove uniqueness, up to constant phase and time translation, within the quantitative single-profile asymptotic class used in the construction. The mechanism is the exponentially small interaction of the profile with its periodic images. Exact mass and energy constraints select the concentrating branch, while a positive energy on the moving constrained space controls the terminal-value problem at every fixed regularity order.