Large time behavior for the cubic nonlinear Schrödinger equation

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension (1) {iu_t + 1/2u_xx + u^-3 = 0, t ∈ R, x ∈ R, u(0, x) = u₀(x), x ∈ R.} Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data u₀ ∈ H^1,0 ∩ H^0,1 are small and such that sup_{|ξ| ≤ 1} | arg ℱu₀(ξ) - πn/2 | < π/8 for some n ∈ Z, and inf_{|ξ| ≤ 1} |ℱu₀(ξ)| > 0, then the solution has an additional logarithmic time-decay in the short range region |x| ≤ √t. In the far region |x| > √t the asymptotics have a quasilinear character.