Type: Article

Logarithmic Time Decay for the Cubic Nonlinear Schrödinger Equations

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We prove global existence of solutions to the Cauchy problem for the 1D cubic nonlinear Schrödinger equation iu t +12 u xx =?e i?2 u 3 +|u| 2 u,x?R, t>1, (0.1)where ??R, 0<|?|<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|2u in Equation (0.1).