L2-decay rate for the critical nonlinear Schrödinger equation with a small smooth data

We consider the Cauchy problem for the one dimensional nonlinear dissipative Schrödinger equation with a cubic nonlinearity λ | u | 2 u , where λ ∈ C with Im λ < 0 . We show that a relation between L 2 -decay rate for the solution and a smoothness of the initial data. Our result improves the recent work of Hayashi–Li–Naumkin (Adv Math Phys Art. ID 3702738, 7, 2016) for the decay rate of L 2 .