Logarithmic Time Decay for the Cubic Nonlinear Schrödinger Equations

We prove global existence of solutions to the Cauchy problem for the 1D cubic nonlinear Schrödinger equation iut + 1/2 u_xx =λ e^iπ/2 u³ + |u|²u, x ε R, t>1, (0.1) where λ ε R, 0<|λ|< √ 3.We show that the time decay estimate of the solution in the far region |x| > √ t coincides with that for the linear case, whereas in the short-range region |x| ≤ √ t the solution obtains an extra logarithmic time decay, which is less than that in the absence of the resonance term |u|2uin Equation (0.1).